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Chapter 3: Problem 54

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$4 \log (x-6)=11$$

Short Answer

Expert verified
The solution to the equation is approximately \(x \approx 35.707\)

Step by step solution

01

Divide both sides of the equation by 4

We can simplify the equation by dividing both sides by 4, giving us \(\log (x-6) = \frac{11}{4}\)
02

Convert to exponential form

The equation \(\log_a b = c\) can be converted to exponential form as \(a^c = b\), applying this to our equation gives \(10^{\frac{11}{4}} = x-6\).
03

Solve for x

We can solve for \(x\) by adding 6 to both sides of the equation to isolate \(x\), giving \(x = 10^{\frac{11}{4}} + 6\)
04

Approximate the solution

We can use a calculator to evaluate \(10^{\frac{11}{4}} + 6\) and round to 3 decimal places, if necessary.

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Most popular questions from this chapter

In Exercises \(103-106,\) use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$

Evaluate \(g(x)=\ln x\) at the indicated value of \(x\) without using a calculator. $$x=e^{-4}$$

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$2+3 \ln x=12$$

Using the One-to-One Property In Exercises \(73-76,\) use the One-to-One Property to solve the equation for \(x\). \(\ln \left(x^{2}-x\right)=\ln 6\)

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$3 \ln 5 x=10$$
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