Problem 46

Question

Write each exponential equation as a logarithmic equation. See Example 2. $$ m^{n}=p $$

Step-by-Step Solution

Verified

Answer

$ n = \log_{m}(p) $

1Step 1: Understanding the Exponential Equation

The given exponential equation is $ m^{n} = p $. In this equation, $ m $ is the base, $ n $ is the exponent, and $ p $ is the result.

2Step 2: Writing the Corresponding Logarithmic Form

To convert the exponential equation to its logarithmic form, use the relationship between exponents and logarithms. The general form is: if $ a^{b} = c $, then the logarithmic form is $ b = \log_{a}(c) $. Apply this to the given equation.

3Step 3: Applying the Formula

For the equation $ m^{n} = p $, the base $ m $ becomes the base of the logarithm, $ n $ is the output of the log equation, and $ p $ is the input to the logarithm. Therefore, the logarithmic form is $ n = \log_{m}(p) $.

Key Concepts

Exponential EquationsConverting EquationsLogarithmic Form

Exponential Equations

Exponential equations are a fundamental aspect of algebra that involve variables in the exponent. In an exponential equation, such as $ m^n = p $:

$ m $ is the base, which is a constant value.
$ n $ is the exponent, indicating how many times the base is multiplied by itself.
$ p $ is the result of the exponentiation.

Exponential equations are used to model situations involving growth or decay, such as population growth, radioactive decay, or compound interest.
Understanding how these equations work lays the groundwork for converting them into other forms, like logarithmic equations, which can be more convenient for solving certain types of problems.

Converting Equations

Converting equations between different forms is an essential skill in algebra. Specifically, moving from exponential form to logarithmic form helps in analyzing and solving equations more easily. To convert an exponential equation like $ a^b = c $ into a logarithmic equation, we rely on the relationship between exponentials and logarithms.

The base of the exponential equation becomes the base of the logarithm.
The exponent is the equivalent of the result in the logarithmic form.
The result of the exponential equation becomes the input value for the logarithmic expression.

This conversion is based on the formula: if $ a^b = c $, then $ b = \log_a(c) $.
By converting equations, we can tackle problems using logarithmic properties, which often simplify solving for unknowns.

Logarithmic Form

Logarithms are mathematical functions that help determine the power to which a base must be raised to obtain a particular number. The logarithmic form is derived through conversion from an exponential equation, aiding in solving exponential problems.
For an equation like $ m^n = p $, once converted, we get $ n = \log_m(p) $. Here's what the components represent:

$ n $ is the exponent from the original equation, now the result of the logarithm.
$ m $, the base of the exponent, is also the base of the log.
$ p $ is the number we are "taking the log of" to find $ n $.

The logarithmic form exposes the inverse nature of exponents and logs, making it an invaluable tool for solving equations where the variable is an exponent.
Knowing how to write equations in logarithmic form facilitates understanding and solving complex algebraic expressions.

Problem 46

Other exercises in this chapter

Problem 46

Write each logarithm as the sum and/or difference of logarithms of a single quantity. Then simplify, if possible. See Example 4. $$ \log _{2} \frac{a b}{4} $$

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Problem 46

Solve each equation. $$ \log (2-x)=3 $$

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Problem 46

Each of the following functions is one-to-one. Find the inverse of each function and express it using $f^{-1}(x)$ notation. $$ f(x)=x^{3}-4 $$

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Problem 47

Evaluate each expression without using a calculator. $$ \ln \sqrt[4]{e} $$

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