Solve for x 3^(4x-5)<8

$3^{4 x - 5} < 8$

Step 1

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

$\ln (3^{4 x - 5}) < \ln (8)$

Step 2

Expand $\ln (3^{4 x - 5})$ by moving $4 x - 5$ outside the logarithm.

$(4 x - 5) \ln (3) < \ln (8)$

Step 3

Simplify the left side.

Apply the distributive property.

$4 x \ln (3) - 5 \ln (3) < \ln (8)$

Step 4

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

$4 x \ln (3) - 5 \ln (3) - \ln (8) = 0$

Step 5

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

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

$4 x \ln (3) - \ln (8) = 5 \ln (3)$

Add $\ln (8)$ to both sides of the equation.

$4 x \ln (3) = 5 \ln (3) + \ln (8)$

Step 6

Divide each term in $4 x \ln (3) = 5 \ln (3) + \ln (8)$ by $4 \ln (3)$ and simplify.

Divide each term in $4 x \ln (3) = 5 \ln (3) + \ln (8)$ by $4 \ln (3)$ .

$\frac{4 x \ln (3)}{4 \ln (3)} = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x \ln (3)}{4 \ln (3)} = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Rewrite the expression.

$\frac{x \ln (3)}{\ln (3)} = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$\frac{x \ln (3)}{\ln (3)} = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Divide $x$ by $1$ .

$x = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Simplify the right side.

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

$x = \frac{5 \ln (3)}{4 \ln (3)} + \frac{\ln (8)}{4 \ln (3)}$

Rewrite the expression.

$x = \frac{5}{4} + \frac{\ln (8)}{4 \ln (3)}$

Step 7

The solution consists of all of the true intervals.

$x < \frac{5}{4} + \frac{\ln (8)}{4 \ln (3)}$

Step 8

The result can be shown in multiple forms.

Inequality Form:

$x < \frac{5}{4} + \frac{\ln (8)}{4 \ln (3)}$

Interval Notation:

$(- \infty, \frac{5}{4} + \frac{\ln (8)}{4 \ln (3)})$

Step 9

Solve for x 3^(4x-5)<8

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