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Chapter 4: Problem 10

Write each equation in its equivalent logarithmic form. $$5^{4}=625$$

Short Answer

Expert verified
\(\log_{5}625 = 4\)

Step by step solution

01

STEP 1: Understand the Concept

Exponential and logarithmic functions are inverses of each other. Here's how we can write an exponential function as a logarithmic function and vice-versa: if \(a = b^{c}\), then the equivalent logarithmic form is \(log_b a = c\). Essentially central numbers stay the same, while outer numbers switch positions.
02

STEP 2: Write equation in exponential form

The given equation \(5^{4}=625\) is already in exponential form where \(b = 5\), \(c = 4\) and \(a = 625\). Next, employ the relationship explained in Step 1 to convert this to logarithm form.
03

STEP 3: Write equation in log form

Applying the definition of logarithm, the equation transforms as \(\log_{5}625 = 4\). In other words, it can be read as: 'log to base 5 of 625 is equal to 4', which essentially means that the number 5 must be raised to the power of 4 to get 625.

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Most popular questions from this chapter

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \sqrt[5]{x} $$

Rescarch applications of logarithmic functions as mathematical models and plan a seminar based on your group's research. Each group member should research one of the following areas or any other area of interest: \(\mathrm{pH}\) (acidity of solutions), intensity of sound (decibels), brightness of stars, consumption of natural resources, human memory, progress over time in a sport, profit over time. For the area that you select. explain how logarithmic functions are used and provide examples.

The formula $$ t=\frac{1}{c}[\ln A-\ln (A-N)] $$ describes the time, \(t,\) in weeks, that it takes to achieve mastery of a portion of a task, where \(A\) is the maximum learning possible, \(N\) is the portion of the learning that is to be achieved, and \(c\) is a constant used to measure an individual's learning style. a. Express the formula so that the expression in brackets is written as a single logarithm. b. The formula is also used to determine how long it will take chimpanzees and apes to master a task. For example, a typical chimpanzce learning sign language can master a maximum of 65 signs. Use the form of the formula from part (a) to answer this question: How many weeks will it take a chimpanzee to master 30 signs if \(c\) for that chimp is \(0.03 ?\)

Without showing the details, explain how to condense \(\ln x-2 \ln (x+1)\)

In Exercises \(41-70\), use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is \(I .\) Where possible, evaluate logarithmic expressions. $$ \ln x+\ln 7 $$
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