Question

PLEASE HELP
Explain how to solve 5x^(− 2) = 8 using the change of base formula log base b of y equals log y over log b. Include the solution for x in your answer. Round your answer to the nearest thousandth.

Answer 1

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Answer:

  x ≈ 0.791

Step-by-step explanation:

You can divide by 5 to get the exponential factor by itself.

  x^(-2) = 8/5 = 1.6

Taking logarithms base x gives ...

  $\log_x{1.6}=-2$

Using the change of base formula, this can be rewritten to facilitate solution for x.

  $\dfrac{\log{1.6}}{\log{x}}=-2\\\\ \log{x}=-\dfrac{\log{1.6}}{2}=\log{1.6^{-\frac{1}{2}}}\\\\x=1.6^{-\frac{1}{2}}=\dfrac{1}{\sqrt{1.6}}=\dfrac{\sqrt{10}}{4}\\\\x\approx 0.791$

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Alternate solution

The equation can be solved without using logarithms at all.

  x^-2 = 8/5 . . . . divide by 5

  x^2 = 5/8 . . . . . invert both sides

  x = √(5/8) = √(10/16) = (√10)/4