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Chapter 4: Problem 23

Find the indicated value of the logarithmic functions. $$\log _{1 / 5}(125)$$

Short Answer

Expert verified
-3

Step by step solution

01

Understand the Logarithmic Form

Recall that \(\text{if } \log_b(a) = c, \text{ then } b^c = a\). In this problem, we need to find \(\log_{1/5}(125)\). This implies \(\left(\frac{1}{5}\right)^c = 125\).
02

Rewrite the Base and Argument

Rewrite the fraction \(\frac{1}{5}\) as \(5^{-1}\). So the equation becomes \(\left(5^{-1}\right)^c = 125\). Simplify the left side: \(5^{-c} = 125\).
03

Express 125 as a Power of 5

Determine the power of 5 that equals 125: \(125 = 5^3\). Therefore, the equation now is \(5^{-c} = 5^3\).
04

Set the Exponents Equal

Since the bases are the same, set the exponents equal to each other: \(-c = 3\).
05

Solve for c

Solve the equation for \(c\) by multiplying both sides by \(-1\): \c = -3\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithms
Logarithms are the inverse operation to exponentiation. They help us solve for the unknown exponent in an equation of the form \(b^c = a\). Here, \log_b(a) = c\ indicates that \(b\) raised to the power \(c\) equals \(a\). Understanding logarithms is essential because they transform multiplicative processes into additive ones, making complex computations more manageable. For example, in the given problem, you need to identify the value of the logarithmic function \log_{1/5}(125)\ by converting it into an equation involving exponentiation.
Exponentiation
Exponentiation is a mathematical operation where a number (the base) is raised to a certain power (the exponent). For example, \(5^3 = 125\). This operation is the fundamental process reversed by logarithms. In the context of the problem \log_{1/5}(125)\, converting the logarithm to its exponential form helps in solving it. \left(\frac{1}{5}\right)^c = 125\. This reveals that we are trying to find an exponent \(c\) such that \(\frac{1}{5}\) raised to that power equals 125.
Base Conversion
Base conversion involves changing the representation of a number from one base to another. For instance, converting the base \( \frac{1}{5} \) to \5^{-1}\ is a crucial step in solving the given problem. By rewriting \left(\frac{1}{5}\right)^c\ as \left(5^{-1}\right)^c\, it simplifies to \5^{-c}\. This makes it straightforward to compare the exponents once the bases match. Hence, understanding base conversion allows us to handle and manipulate complex expressions with ease.
Solving Equations
Solving logarithmic equations often involves converting them to an exponential form and then isolating the unknown variable. For \log_{1/5}(125)\, following these steps simplifies the process:
  • Rewrite \frac{1}{5}\ as \5^{-1}\ to obtain \5^{-c} = 125\.
  • Then, express 125 as \( 5^3 \).
With the same base on both sides, you can set the exponents equal to each other: \-c = 3\.
Finally, solving for \c\ by multiplying both sides by \-1\ yields \c = -3\. These steps show how converting forms and isolating variables help solve logarithmic problems effectively.

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Most popular questions from this chapter

Solve each problem. When needed, use 365 days per year and 30 days per month. Periodic Compounding Melinda invests her \(\$ 80,000\) winnings from Publishers Clearing House at a \(9 \%\) annual percentage rate. Find the amount of the investment at the end of 20 years and the amount of interest earned during the 20 years if the interest is compounded a. annually b. quarterly c. monthly d. daily.

Solve each equation. Find the exact solutions. $$\log _{2}\left(\log _{2}\left(\log _{2}\left(2^{\left(4^{\circ}\right)}\right)\right)\right)=3$$

Solve \(\frac{3}{8}(x-5)=\frac{3}{2} x+\frac{5}{6}.\)

Solve each problem. The population of the world doubled from 1950 to \(1987,\) going from 2.5 billion to 5 billion people. Using the exponential model, $$P=P_{0} e^{r t},$$ find the annual growth rate \(r\) for that period. Although the annual growth rate has declined slightly to \(1.63 \%\) annually, the population of the world is still growing at a tremendous rate. Using the initial population of 5 billion in 1987 and an annual rate of \(1.63 \%\), estimate the world population in the year 2010

Two-Parent Families The percentage of households with children that consist of two-parent families is shown in the following table (U.S. Census Bureau, www.census.gov). $$\begin{array}{|c|c|} \hline \text { Year } & \text { Two Parents } \\ \hline 1970 & 85 \% \\ 1980 & 77 \\ 1990 & 73 \\ 1992 & 71 \\ 1994 & 69 \\ 1996 & 68 \\ 1998 & 68 \\ 2000 & 68 \\ \hline \end{array}$$ a. Use logarithmic regression on a graphing calculator to find the best- fitting curve of the form \(y=a+b \cdot \ln (x)\) where \(x=0\) corresponds to 1960. b. Use your equation to predict the percentage of two-parent families in 2010 . c. In what year will the percentage of two-parent families reach \(50 \% ?\). d. Graph your equation and the data on your graphing calculator. Does this logarithmic model look like another model that we have used?
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