Evaluate Using the Given Value 4^(1+2x)=1 , 24

$4^{(1 + 2 x)} = 1$ , $24$

Step 1

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

$\ln (4^{1 + 2 x}) = \ln (1)$

Step 2

Expand $\ln (4^{1 + 2 x})$ by moving $1 + 2 x$ outside the logarithm.

$(1 + 2 x) \ln (4) = \ln (1)$

Step 3

Simplify the left side.

Simplify $(1 + 2 x) \ln (4)$ .

Apply the distributive property.

$1 \ln (4) + 2 x \ln (4) = \ln (1)$

Multiply $\ln (4)$ by $1$ .

$\ln (4) + 2 x \ln (4) = \ln (1)$

Step 4

Simplify the right side.

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

$\ln (4) + 2 x \ln (4) = 0$

Step 5

Reorder $\ln (4)$ and $2 x \ln (4)$ .

$2 x \ln (4) + \ln (4) = 0$

Step 6

Subtract $\ln (4)$ from both sides of the equation.

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

Step 7

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

Divide each term in $2 x \ln (4) = - \ln (4)$ by $2 \ln (4)$ .

$\frac{2 x \ln (4)}{2 \ln (4)} = \frac{- \ln (4)}{2 \ln (4)}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x \ln (4)}{2 \ln (4)} = \frac{- \ln (4)}{2 \ln (4)}$

Rewrite the expression.

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (4)}{2 \ln (4)}$

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$\frac{x \ln (4)}{\ln (4)} = \frac{- \ln (4)}{2 \ln (4)}$

Divide $x$ by $1$ .

$x = \frac{- \ln (4)}{2 \ln (4)}$

Simplify the right side.

Cancel the common factor of $\ln (4)$ .

Cancel the common factor.

$x = \frac{- \ln (4)}{2 \ln (4)}$

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 8

Replace the variable $x$ with $- \frac{1}{2}$ in the expression.

$24$

Evaluate Using the Given Value 4^(1+2x)=1 , 24

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