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Chapter 4: Problem 53

Solve each logarithmic equation. Be sure to reject any value of \(x\) that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. $$\log _{4}(x+5)=3$$

Short Answer

Expert verified
The exact solution to the logarithmic equation \(\log _{4}(x+5)=3\) is \(x=59\).

Step by step solution

01

Transform the equation from logarithmic to exponential form

We know from the properties of logarithms that \(\log _{b}a=c\) is equivalent to \(b^c=a\). So, we can write the given equation as \(4^3=x+5\). Now the equation is in a form we can solve for \(x\).
02

Solve the equation for \(x\)

Evaluating \(4^3\), we get 64. So, the equation becomes \(64=x+5\). By isolating \(x\) on one side of the equation, the solution becomes \(x=64-5\). Hence, \(x=59\).
03

Check the value of \(x\) to ensure it's in the domain of the original expression

The original expression is \(\log _{4}(x+5)\) which implies for any value of \(x, x+5 >0\). Plugging in our value of \(x=59\) into the equation, we get \(59+5=64\), which is greater than 0. Therefore, \(x=59\) is in the domain of the original logarithmic expressions.

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Most popular questions from this chapter

Use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution=set. Verify this value by direct substitution into the equation. $$5^{x}=3 x+4$$

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Find the domain of each logarithmic function. $$f(x)=\log \left(\frac{x-2}{x+5}\right)$$
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