Question

1. use logarithms to solve the exponential equation.

$4^{1 - 8x}=5^{x + 5}$
$x = \square$ (round to three decimal places as needed.)

Explanation:

Step1: Take log of both sides

$\log(4^{1-8x}) = \log(5^{x+5})$

Step2: Apply power rule of logs

$(1-8x)\log(4) = (x+5)\log(5)$

Step3: Expand both sides

$\log(4) - 8x\log(4) = x\log(5) + 5\log(5)$

Step4: Isolate x terms

$\log(4) - 5\log(5) = x\log(5) + 8x\log(4)$

Step5: Factor out x on right

$\log(4) - 5\log(5) = x(\log(5) + 8\log(4))$

Step6: Solve for x

$x = \frac{\log(4) - 5\log(5)}{\log(5) + 8\log(4)}$

Step7: Calculate numerical value

Using $\log(4)\approx0.6021$, $\log(5)\approx0.6990$:
$x = \frac{0.6021 - 5(0.6990)}{0.6990 + 8(0.6021)} = \frac{0.6021 - 3.495}{0.6990 + 4.8168} = \frac{-2.8929}{5.5158}\approx-0.525$

Answer:

$-0.525$