Solve for x 2^(3x-4)=5

$2^{3 x - 4} = 5$

Step 1

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

$\ln (2^{3 x - 4}) = \ln (5)$

Step 2

Expand $\ln (2^{3 x - 4})$ by moving $3 x - 4$ outside the logarithm.

$(3 x - 4) \ln (2) = \ln (5)$

Step 3

Simplify the left side.

Apply the distributive property.

$3 x \ln (2) - 4 \ln (2) = \ln (5)$

Step 4

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

$3 x \ln (2) - 4 \ln (2) - \ln (5) = 0$

Step 5

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

Add $4 \ln (2)$ to both sides of the equation.

$3 x \ln (2) - \ln (5) = 4 \ln (2)$

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

$3 x \ln (2) = 4 \ln (2) + \ln (5)$

Step 6

Divide each term in $3 x \ln (2) = 4 \ln (2) + \ln (5)$ by $3 \ln (2)$ and simplify.

Divide each term in $3 x \ln (2) = 4 \ln (2) + \ln (5)$ by $3 \ln (2)$ .

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x \ln (2)}{3 \ln (2)} = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

Rewrite the expression.

$\frac{x \ln (2)}{\ln (2)} = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

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

Cancel the common factor.

$\frac{x \ln (2)}{\ln (2)} = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

Divide $x$ by $1$ .

$x = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

Simplify the right side.

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

Cancel the common factor.

$x = \frac{4 \ln (2)}{3 \ln (2)} + \frac{\ln (5)}{3 \ln (2)}$

Rewrite the expression.

$x = \frac{4}{3} + \frac{\ln (5)}{3 \ln (2)}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{3} + \frac{\ln (5)}{3 \ln (2)}$

Decimal Form:

$x = 2.10730936 \dots$

Solve for x 2^(3x-4)=5

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