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Chapter 4: Problem 139

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I can solve \(4^{x}=15\) by writing the equation in logarithmic form.

Short Answer

Expert verified
The statement makes sense. The equation \(4^{x}=15\) can be converted into logarithmic form as \( \log_{4}{15}=x\) to make the solution process simpler.

Step by step solution

01

Understanding the statement

The statement says, 'I can solve \(4^{x}=15\) by writing the equation in logarithmic form.' This will be examined for the validity and reasoning behind it.
02

Changing the equation to logarithmic form

The equation \(4^{x}=15\) can be converted to logarithmic form by applying the definition of logarithms. In this case, it will be written as \( \log_{4}{15}=x\). This means that 'x is the power to which 4 must be raised to get 15'.
03

Evaluating the reasonability of the solution

Once the equation has been converted to logarithmic form, solving for x becomes more straightforward. This is because the x term that was previously in the exponent position is now isolated on one side of the equation, making it easier to solve. Therefore, the initial statement makes sense and it is indeed easier to solve the equation by converting it into logarithmic form.

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Most popular questions from this chapter

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. $$ \text { If } \log (x+3)=2, \text { then } e^{2}=x+3 $$

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. $$ 3^{x}=2 x+3 $$

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. $$ 3^{x+1}=9 $$

The exponential growth models describe the population of the indicated country, \(A,\) in millions, t years after 2006 . $$ \begin{array}{ll} {\text { Canada }} & {A=33.1 e^{0.009 t}} \\ {\text { Uganda }} & {A=28.2 e^{0.034 t}} \end{array} $$ Use this information to determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The models indicate that in 2013 , Uganda's population will exceed Canada's.

Exercises 150鈥�152 will help you prepare for the material covered in the next section. In each exercise, evaluate the indicated logarithmic expressions without using a calculator. a. Evaluate: \(\log _{3} 81\) b. Evaluate: \(2 \log _{3} 9\) c. What can you conclude about $$ \log _{3} 81, \text { or } \log _{3} 9^{2} ? $$
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