Solve for x 5^(2x-3)=1

$5^{2 x - 3} = 1$

Step 1

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

$\ln (5^{2 x - 3}) = \ln (1)$

Step 2

Expand $\ln (5^{2 x - 3})$ by moving $2 x - 3$ outside the logarithm.

$(2 x - 3) \ln (5) = \ln (1)$

Step 3

Simplify the left side.

Apply the distributive property.

$2 x \ln (5) - 3 \ln (5) = \ln (1)$

Step 4

Simplify the right side.

The natural logarithm of $1$ is $0$ .

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

Step 5

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

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

Step 6

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

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

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

Cancel the common factor.

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

Rewrite the expression.

$x = \frac{3}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3}{2}$

Decimal Form:

$x = 1.5$

Mixed Number Form:

$x = 1 \frac{1}{2}$

Solve for x 5^(2x-3)=1

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