Solve for x 278^(4x)=37800

$278^{4 x} = 37800$

Step 1

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

$\ln (278^{4 x}) = \ln (37800)$

Step 2

Expand $\ln (278^{4 x})$ by moving $4 x$ outside the logarithm.

$4 x \ln (278) = \ln (37800)$

Step 3

Divide each term in $4 x \ln (278) = \ln (37800)$ by $4 \ln (278)$ and simplify.

Divide each term in $4 x \ln (278) = \ln (37800)$ by $4 \ln (278)$ .

$\frac{4 x \ln (278)}{4 \ln (278)} = \frac{\ln (37800)}{4 \ln (278)}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x \ln (278)}{4 \ln (278)} = \frac{\ln (37800)}{4 \ln (278)}$

Rewrite the expression.

$\frac{x \ln (278)}{\ln (278)} = \frac{\ln (37800)}{4 \ln (278)}$

Cancel the common factor of $\ln (278)$ .

Cancel the common factor.

$\frac{x \ln (278)}{\ln (278)} = \frac{\ln (37800)}{4 \ln (278)}$

Divide $x$ by $1$ .

$x = \frac{\ln (37800)}{4 \ln (278)}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{\ln (37800)}{4 \ln (278)}$

Decimal Form:

$x = 0.46822912 \dots$