/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 54 Solve the following equations. ... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 54

Solve the following equations. $$\log _{b} 125=3$$

Short Answer

Expert verified
Answer: The value of \(b\) is 5.

Step by step solution

01

Identify the logarithmic equation

Given equation is \(\log _{b} 125 = 3\).
02

Convert the logarithmic equation into the exponential form

Recall the rule for logarithm to exponent conversion: $$\log_{a} b = c \Rightarrow a^{c} = b.$$ Using the given logarithmic equation \(\log _{b} 125=3\), we can convert it into an exponential form: $$b^3 = 125.$$
03

Solve for \(b\)

Now we have \(b^3 = 125\). To find the value of \(b\), we can take the cube root of both sides: $$\sqrt[3]{b^3} = \sqrt[3]{125},$$ which simplifies to $$b = 5.$$ Thus, the value of \(b\) is 5.

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Key Concepts

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

Logarithm to Exponent Conversion
Understanding the relationship between logarithms and exponents is crucial for solving logarithmic equations effectively. When facing a logarithmic equation like \(\log _{b} 125=3\), the first step is to convert this logarithm to its equivalent exponential form.

The general rule for converting a logarithm to an exponent is as follows: If you have an equation \(\log_{a} b = c\), it can be rewritten as \(a^{c} = b\). This is based on the definition of a logarithm, which tells us that if a raised to the power of c equals b, then the logarithm of b with base a is c.

Apply this rule to our problem, and you transform the logarithmic equation from \(\log _{b} 125=3\) to \(b^3 = 125\). This conversion is a pivotal step in finding the value of 'b'.
Exponential Form
Once the logarithm to exponent conversion is made, we work with the equation in its exponential form. An equation is said to be in exponential form when a variable is expressed as a base raised to an exponent. In our case, \(b^3 = 125\) is an exponential equation, indicating that 'b' raised to the power of 3 equals 125.

The goal here is to solve for the base 'b'. Exponential equations are often easier to manipulate and solve because they are more direct and allow for the use of root functions to isolate the variable. In this situation, identifying that '125' is a perfect cube aids in swiftly solving for 'b' through the application of the cube root, which is the inverse operation to cubing a number.
Cube Root
The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 125 is the number that, when cubed, equals 125. Symbolically, it's written as \(\sqrt[3]{125}\).

To solve the exponential equation \(b^3 = 125\), we apply the cube root to both sides, which looks like \(\sqrt[3]{b^3} = \sqrt[3]{125}\). Since the cube root of 125 is 5 (because \(5\times 5\times 5 = 125\)), we find that 'b' must be 5.

In solving logarithmic equations, recognizing which root function to apply is key. By employing the cube root in this step, we're able to directly determine the value of 'b' and reach a succinct and accurate conclusion to our equation.

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Most popular questions from this chapter

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