Algebra Examples

$x^{5} \cdot x^{4} = a x^{z}$

Step 1

Solve the equation for $z$ .

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Step 1.1

Rewrite the equation as $a x^{z} = x^{5} \cdot x^{4}$ .

$a x^{z} = x^{5} \cdot x^{4}$

Step 1.2

Multiply $x^{5}$ by $x^{4}$ by adding the exponents.

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Step 1.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$a x^{z} = x^{5 + 4}$

Step 1.2.2

Add $5$ and $4$ .

$a x^{z} = x^{9}$

Step 1.3

Divide each term in $a x^{z} = x^{9}$ by $a$ and simplify.

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Step 1.3.1

Divide each term in $a x^{z} = x^{9}$ by $a$ .

$\frac{a x^{z}}{a} = \frac{x^{9}}{a}$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Cancel the common factor of $a$ .

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Step 1.3.2.1.1

Cancel the common factor.

$\frac{a x^{z}}{a} = \frac{x^{9}}{a}$

Step 1.3.2.1.2

Divide $x^{z}$ by $1$ .

$x^{z} = \frac{x^{9}}{a}$

Step 1.4

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

$\ln (x^{z}) = \ln (\frac{x^{9}}{a})$

Step 1.5

Expand $\ln (x^{z})$ by moving $z$ outside the logarithm.

$z \ln (x) = \ln (\frac{x^{9}}{a})$

Step 1.6

Divide each term in $z \ln (x) = \ln (\frac{x^{9}}{a})$ by $\ln (x)$ and simplify.

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Step 1.6.1

Divide each term in $z \ln (x) = \ln (\frac{x^{9}}{a})$ by $\ln (x)$ .

$\frac{z \ln (x)}{\ln (x)} = \frac{\ln (\frac{x^{9}}{a})}{\ln (x)}$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $\ln (x)$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{z \ln (x)}{\ln (x)} = \frac{\ln (\frac{x^{9}}{a})}{\ln (x)}$

Step 1.6.2.1.2

Divide $z$ by $1$ .

$z = \frac{\ln (\frac{x^{9}}{a})}{\ln (x)}$

Step 2

Simplify the right side.

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Step 2.1

Simplify $\frac{\ln (\frac{{(a x^{z})}^{9}}{a})}{\ln (a x^{z})}$ .

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Step 2.1.1

Simplify the numerator.

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Step 2.1.1.1

Apply the product rule to $a x^{z}$ .

$z = \frac{\ln (\frac{a^{9} {(x^{z})}^{9}}{a})}{\ln (a x^{z})}$

Step 2.1.1.2

Multiply the exponents in ${(x^{z})}^{9}$ .

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Step 2.1.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$z = \frac{\ln (\frac{a^{9} x^{z \cdot 9}}{a})}{\ln (a x^{z})}$

Step 2.1.1.2.2

Move $9$ to the left of $z$ .

$z = \frac{\ln (\frac{a^{9} x^{9 z}}{a})}{\ln (a x^{z})}$

Step 2.1.2

Cancel the common factor of $a^{9}$ and $a$ .

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Step 2.1.2.1

Factor $a$ out of $a^{9} x^{9 z}$ .

$z = \frac{\ln (\frac{a (a^{8} x^{9 z})}{a})}{\ln (a x^{z})}$

Step 2.1.2.2

Cancel the common factors.

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Step 2.1.2.2.1

Raise $a$ to the power of $1$ .

$z = \frac{\ln (\frac{a (a^{8} x^{9 z})}{a^{1}})}{\ln (a x^{z})}$

Step 2.1.2.2.2

Factor $a$ out of $a^{1}$ .

$z = \frac{\ln (\frac{a (a^{8} x^{9 z})}{a \cdot 1})}{\ln (a x^{z})}$

Step 2.1.2.2.3

Cancel the common factor.

$z = \frac{\ln (\frac{a (a^{8} x^{9 z})}{a \cdot 1})}{\ln (a x^{z})}$

Step 2.1.2.2.4

Rewrite the expression.

$z = \frac{\ln (\frac{a^{8} x^{9 z}}{1})}{\ln (a x^{z})}$

Step 2.1.2.2.5

Divide $a^{8} x^{9 z}$ by $1$ .

$z = \frac{\ln (a^{8} x^{9 z})}{\ln (a x^{z})}$