Calculus Examples

$2^{x - 5} = 3$

Step 1

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

$\ln (2^{x - 5}) = \ln (3)$

Step 2

Expand $\ln (2^{x - 5})$ by moving $x - 5$ outside the logarithm.

$(x - 5) \ln (2) = \ln (3)$

Step 3

Simplify the left side.

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Step 3.1

Apply the distributive property.

$x \ln (2) - 5 \ln (2) = \ln (3)$

Step 4

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

$x \ln (2) - 5 \ln (2) - \ln (3) = 0$

Step 5

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

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Step 5.1

Add $5 \ln (2)$ to both sides of the equation.

$x \ln (2) - \ln (3) = 5 \ln (2)$

Step 5.2

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

$x \ln (2) = 5 \ln (2) + \ln (3)$

Step 6

Divide each term in $x \ln (2) = 5 \ln (2) + \ln (3)$ by $\ln (2)$ and simplify.

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Step 6.1

Divide each term in $x \ln (2) = 5 \ln (2) + \ln (3)$ by $\ln (2)$ .

$\frac{x \ln (2)}{\ln (2)} = \frac{5 \ln (2)}{\ln (2)} + \frac{\ln (3)}{\ln (2)}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $\ln (2)$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{5 \ln (2)}{\ln (2)} + \frac{\ln (3)}{\ln (2)}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{5 \ln (2)}{\ln (2)} + \frac{\ln (3)}{\ln (2)}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $\ln (2)$ .

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Step 6.3.1.1

Cancel the common factor.

$x = \frac{5 \ln (2)}{\ln (2)} + \frac{\ln (3)}{\ln (2)}$

Step 6.3.1.2

Divide $5$ by $1$ .

$x = 5 + \frac{\ln (3)}{\ln (2)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = 5 + \frac{\ln (3)}{\ln (2)}$

Decimal Form:

$x = 6.58496250 \dots$