Math  /  Algebra

QuestionConsider the equation 16106x=80-16 \cdot 10^{6 x}=-80 Solve the equation for xx. Express the solution as a logarithm in base-10. \square Approximate the value of xx. Round your answer to the nearest thousandth. xx \approx \square

Studdy Solution

STEP 1

1. The equation 16106x=80-16 \cdot 10^{6x} = -80 is an exponential equation.
2. We need to isolate the exponential term and use logarithms to solve for xx.
3. The solution will be expressed as a logarithm in base-10.

STEP 2

1. Isolate the exponential term 106x10^{6x}.
2. Use logarithms to solve for xx.
3. Approximate the value of xx and round to the nearest thousandth.

STEP 3

First, divide both sides of the equation by 16-16 to isolate the exponential term:
16106x=80 -16 \cdot 10^{6x} = -80
106x=8016 10^{6x} = \frac{-80}{-16}
106x=5 10^{6x} = 5

STEP 4

Take the logarithm of both sides to solve for xx. We will use the base-10 logarithm:
log10(106x)=log10(5) \log_{10}(10^{6x}) = \log_{10}(5)
Using the property of logarithms logb(by)=y\log_{b}(b^y) = y, we have:
6x=log10(5) 6x = \log_{10}(5)

STEP 5

Solve for xx by dividing both sides by 6:
x=log10(5)6 x = \frac{\log_{10}(5)}{6}

STEP 6

Approximate the value of xx using a calculator. Calculate log10(5)\log_{10}(5) and then divide by 6:
log10(5)0.69897 \log_{10}(5) \approx 0.69897
x0.698976 x \approx \frac{0.69897}{6}
x0.116495 x \approx 0.116495
Round to the nearest thousandth:
x0.116 x \approx 0.116
The solution as a logarithm is:
x=log10(5)6 x = \frac{\log_{10}(5)}{6}
The approximate value of xx is:
x0.116 x \approx 0.116

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