Precalculus Examples

$\log (\frac{m}{n}) = \frac{\log (m)}{\log (n)}$

Step 1

Subtract $\frac{\log (m)}{\log (n)}$ from both sides of the equation.

$\log (\frac{m}{n}) - \frac{\log (m)}{\log (n)} = 0$

Step 2

Find the LCD of the terms in the equation.

Tap for more steps...

Step 2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, \log (n), 1$

Step 2.2

The LCM of one and any expression is the expression.

$\log (n)$

Step 3

Multiply each term in $\log (\frac{m}{n}) - \frac{\log (m)}{\log (n)} = 0$ by $\log (n)$ to eliminate the fractions.

Tap for more steps...

Step 3.1

Multiply each term in $\log (\frac{m}{n}) - \frac{\log (m)}{\log (n)} = 0$ by $\log (n)$ .

$\log (\frac{m}{n}) \log (n) - \frac{\log (m)}{\log (n)} \log (n) = 0 \log (n)$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $\log (n)$ .

Tap for more steps...

Step 3.2.1.1

Move the leading negative in $- \frac{\log (m)}{\log (n)}$ into the numerator.

$\log (\frac{m}{n}) \log (n) + \frac{- \log (m)}{\log (n)} \log (n) = 0 \log (n)$

Step 3.2.1.2

Cancel the common factor.

$\log (\frac{m}{n}) \log (n) + \frac{- \log (m)}{\log (n)} \log (n) = 0 \log (n)$

Step 3.2.1.3

Rewrite the expression.

$\log (\frac{m}{n}) \log (n) - \log (m) = 0 \log (n)$

Step 3.3

Simplify the right side.

Tap for more steps...

Step 3.3.1

Multiply $0$ by $\log (n)$ .

$\log (\frac{m}{n}) \log (n) - \log (m) = 0$

Step 4

Solve the equation.

Tap for more steps...

Step 4.1

Rewrite $\log (m) = \log (\frac{m}{n}) \log (n)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$10^{\log (\frac{m}{n}) \log (n)} = m$

Step 4.2

Solve for $m$ .

Tap for more steps...

Step 4.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (10^{\log (\frac{m}{n}) \log (n)}) = \ln (m)$

Step 4.2.2

Expand the left side.

Tap for more steps...

Step 4.2.2.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (10^{(\log (m) - \log (n)) \log (n)}) = \ln (m)$

Step 4.2.2.2

Expand $\ln (10^{(\log (m) - \log (n)) \log (n)})$ by moving $(\log (m) - \log (n)) \log (n)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.3

Simplify the left side.

Tap for more steps...

Step 4.2.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.4

Subtract $\ln (m)$ from both sides of the equation.

$\log (\frac{m}{n}) \log (n) \ln (10) - \ln (m) = 0$

Step 4.2.5

To solve for $m$ , rewrite the equation using properties of logarithms.

$e^{\ln (m)} = e^{\log (\frac{m}{n}) \log (n) \ln (10)}$

Step 4.2.6

Rewrite $\ln (m) = \log (\frac{m}{n}) \log (n) \ln (10)$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7

Solve for $m$ .

Tap for more steps...

Step 4.2.7.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\log (\frac{m}{n}) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.2

Expand the left side.

Tap for more steps...

Step 4.2.7.2.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.2.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.2.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.2.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.3

Simplify the left side.

Tap for more steps...

Step 4.2.7.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.4

Subtract $\ln (m)$ from both sides of the equation.

$\log (\frac{m}{n}) \log (n) \ln (10) - \ln (m) = 0$

Step 4.2.7.5

To solve for $m$ , rewrite the equation using properties of logarithms.

$e^{\ln (m)} = e^{\log (\frac{m}{n}) \log (n) \ln (10)}$

Step 4.2.7.6

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7.7

Solve for $m$ .

Tap for more steps...

Step 4.2.7.7.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\log (\frac{m}{n}) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.2

Expand the left side.

Tap for more steps...

Step 4.2.7.7.2.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.2.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.7.2.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.7.2.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.3

Simplify the left side.

Tap for more steps...

Step 4.2.7.7.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.4

Subtract $\ln (m)$ from both sides of the equation.

$\log (\frac{m}{n}) \log (n) \ln (10) - \ln (m) = 0$

Step 4.2.7.7.5

To solve for $m$ , rewrite the equation using properties of logarithms.

$e^{\ln (m)} = e^{\log (\frac{m}{n}) \log (n) \ln (10)}$

Step 4.2.7.7.6

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7.7.7

Solve for $m$ .

Tap for more steps...

Step 4.2.7.7.7.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\log (\frac{m}{n}) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.2

Expand the left side.

Tap for more steps...

Step 4.2.7.7.7.2.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.2.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.7.7.2.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.7.7.2.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.3

Simplify the left side.

Tap for more steps...

Step 4.2.7.7.7.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.4

Subtract $\ln (m)$ from both sides of the equation.

$\log (\frac{m}{n}) \log (n) \ln (10) - \ln (m) = 0$

Step 4.2.7.7.7.5

To solve for $m$ , rewrite the equation using properties of logarithms.

$e^{\ln (m)} = e^{\log (\frac{m}{n}) \log (n) \ln (10)}$

Step 4.2.7.7.7.6

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7.7.7.7

Solve for $m$ .

Tap for more steps...

Step 4.2.7.7.7.7.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\log (\frac{m}{n}) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.7.2

Expand the left side.

Tap for more steps...

Step 4.2.7.7.7.7.2.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.7.2.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.7.7.7.2.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.7.7.7.2.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.7.3

Simplify the left side.

Tap for more steps...

Step 4.2.7.7.7.7.3.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.7.4

Subtract $\ln (m)$ from both sides of the equation.

$\log (\frac{m}{n}) \log (n) \ln (10) - \ln (m) = 0$

Step 4.2.7.7.7.7.5

To solve for $m$ , rewrite the equation using properties of logarithms.

$e^{\ln (m)} = e^{\log (\frac{m}{n}) \log (n) \ln (10)}$

Step 4.2.7.7.7.7.6

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7.7.7.7.7

Solve for $m$ .

Tap for more steps...

Step 4.2.7.7.7.7.7.1

Subtract $m$ from both sides of the equation.

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} - m = 0$

Step 4.2.7.7.7.7.7.2

Move all the terms containing a logarithm to the left side of the equation.

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m + 0$

Step 4.2.7.7.7.7.7.3

Add $m$ and $0$ .

$e^{\log (\frac{m}{n}) \log (n) \ln (10)} = m$

Step 4.2.7.7.7.7.7.4

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{\log (\frac{m}{n}) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.7.7.5

Expand the left side.

Tap for more steps...

Step 4.2.7.7.7.7.7.5.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.7.7.5.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.7.7.7.7.5.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.7.7.7.7.5.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.7.7.6

Simplify the left side.

Tap for more steps...

Step 4.2.7.7.7.7.7.6.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log (\frac{m}{n}) \log (n) \ln (10) = \ln (m)$

Step 4.2.7.7.7.7.7.7

Expand the left side.

Tap for more steps...

Step 4.2.7.7.7.7.7.7.1

Rewrite $\log (\frac{m}{n})$ as $\log (m) - \log (n)$ .

$\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)}) = \ln (m)$

Step 4.2.7.7.7.7.7.7.2

Expand $\ln (e^{(\log (m) - \log (n)) \log (n) \ln (10)})$ by moving $(\log (m) - \log (n)) \log (n) \ln (10)$ outside the logarithm.

$(\log (m) - \log (n)) \log (n) \ln (10) \ln (e) = \ln (m)$

Step 4.2.7.7.7.7.7.7.3

The natural logarithm of $e$ is $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) \cdot 1 = \ln (m)$

Step 4.2.7.7.7.7.7.7.4

Multiply $\log (m) - \log (n)$ by $1$ .

$(\log (m) - \log (n)) \log (n) \ln (10) = \ln (m)$