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Chapter 4: Problem 122

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Short Answer

Expert verified
The solution for \(x\) in the equation \(3^{x}=140\) is approximated at \(x = \frac{ln(140)}{ln(3)}\), which when calculated gives us the value of \(x\) approximately equal to 4.77.

Step by step solution

01

Identify the exponential equation

The given exponential equation is \(3^{x}=140\). In this equation, 3 is the base, x is the exponent, and 140 is the result of the exponential operation. The goal is to find the value of x.
02

Take the logarithm of both sides

Applying the logarithm to both sides can help us to isolate variable from exponent. We take natural logarithm (ln) of both sides of the equation. This gives us the equation: \(ln(3^{x}) = ln(140)\).
03

Use the properties of logarithms

We use the property of logarithms that allows us to move the exponent in front of the logarithm. This gives us the equation: \(x*ln(3) = ln(140)\).
04

Isolate the variable

We can now find the value of x by dividing both sides of the equation by \(ln(3)\). This gives us the equation: \(x = \frac{ln(140)}{ln(3)}\).
05

Solve the equation

Solving the equation on the calculator provides the value of x.

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Most popular questions from this chapter

Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(1 .\) Where possible, evaluate logarithmic expressions without using a calculator. $$ \log 5+\log 2 $$

The function \(P(x)=95-30 \log _{2} x\) models the percentage, \(P(x),\) of students who could recall the important features of a classroom lecture as a function of time, where \(x\) represents the number of days that have elapsed since the lecture was given. The figure at the top of the next column shows the graph of the function. Use this information to solve Exercises \(117-118\). Round answers to one decimal place. After how many days have all students forgotten the important features of the classroom lecture? (Let \(P(x)=0\) and solve for \(x\).) Locate the point on the graph that conveys this information.

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$ \log _{0.1} 17 $$

Shown, again, in the following table is world population, in billions, for seven selected years from 1950 through \(2010 .\) Using a graphing utility's logistic regression option, we obtain the equation shown on the screen. $$ \begin{array}{cc} {x, \text { Number of Years }} & {y, \text { World Population }} \\ {\text { after } 1949} & {\text { (billions) }} \\ {1(1950)} & {2.6} \\ {11(1960)} & {3.0} \\ {21(1970)} & {3.7} \\ {21(1970)} & {4.5} \\ {41(1990)} & {5.3} \\ {51(2000)} & {6.1} \\ {61(2010)} & {6.9} \end{array} $$ We see from the calculator screen at the bottom of the previous page that a logistic growth model for world population, \(f(x),\) in billions, \(x\) years after 1949 is $$ f(x)=\frac{12.57}{1+4.11 e^{-0.026 x}} $$ Use this function to solve Exercises \(38-42\) How well does the function model the data showing a world population of 6.9 billion for \(2010 ?\)

In Exercises 21鈥�42, evaluate each expression without using a calculator. $$ \log _{7} 49 $$
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