Pre-Algebra Examples

$5^{X - 3} = 10$

Step 1

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

$\ln (5^{X - 3}) = \ln (10)$

Step 2

Expand $\ln (5^{X - 3})$ by moving $X - 3$ outside the logarithm.

$(X - 3) \ln (5) = \ln (10)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$X \ln (5) - 3 \ln (5) = \ln (10)$

Step 4

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

$X \ln (5) - 3 \ln (5) - \ln (10) = 0$

Step 5

Move all terms not containing $X$ to the right side of the equation.

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Step 5.1

Add $3 \ln (5)$ to both sides of the equation.

$X \ln (5) - \ln (10) = 3 \ln (5)$

Step 5.2

Add $\ln (10)$ to both sides of the equation.

$X \ln (5) = 3 \ln (5) + \ln (10)$

Step 6

Divide each term in $X \ln (5) = 3 \ln (5) + \ln (10)$ by $\ln (5)$ and simplify.

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Step 6.1

Divide each term in $X \ln (5) = 3 \ln (5) + \ln (10)$ by $\ln (5)$ .

$\frac{X \ln (5)}{\ln (5)} = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (10)}{\ln (5)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (5)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{X \ln (5)}{\ln (5)} = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (10)}{\ln (5)}$

Step 6.2.1.2

Divide $X$ by $1$ .

$X = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (10)}{\ln (5)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (5)$ .

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Step 6.3.1.1

Cancel the common factor.

$X = \frac{3 \ln (5)}{\ln (5)} + \frac{\ln (10)}{\ln (5)}$

Step 6.3.1.2

Divide $3$ by $1$ .

$X = 3 + \frac{\ln (10)}{\ln (5)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$X = 3 + \frac{\ln (10)}{\ln (5)}$

Decimal Form:

$X = 4.43067655 \dots$