Problem 23 - Precalculus

Question

Write each exponential equation in its equivalent logarithmic form. $$78,125=5^{7}$$

Step-by-Step Solution

Answer

The logarithmic form is \( \log_5(78,125) = 7 \).

Step 1: Understand the format of the given equation

The given equation is in the exponential form, where it is written as \( b^x = y \). Here, \( b \) is the base, \( x \) is the exponent, and \( y \) is the result. In this equation, \( b = 5 \), \( x = 7 \), and \( y = 78,125 \).

Step 2: Recall the definition of logarithms

A logarithm is the inverse operation to exponentiation. The logarithmic form of an exponential equation \( b^x = y \) is \( \log_b(y) = x \). This means that \( x \) is the power to which the base \( b \) must be raised to get \( y \).

Step 3: Rewrite the equation in logarithmic form

Using the rule from the previous step, convert the given exponential equation \( 78,125 = 5^{7} \) to logarithmic form as \( \log_5(78,125) = 7 \).

Key Concepts

Exponential Equations

An exponential equation is a mathematical expression where a constant base is raised to a variable exponent, such as \( b^x = y \). In this format, \( b \) is referred to as the base, \( x \) is the exponent, and \( y \) is the result.
Exponential equations often appear in problems involving growth or decay, like population growth or radioactive decay. These equations help express phenomena that increase or decrease multiplicatively.

• Exponential equations are used across various fields such as finance, science, and engineering.
• They're essential for expressing changes that happen rapidly over time, often involving powers of 2, 10, or any other base.

To solve exponential equations, sometimes it is necessary to use logarithms, especially when the exponent is unknown. Changing the equation to logarithmic form can make the process of solving much more straightforward. Just remember that any power with a base greater than one will always increase as the exponent increases.

Inverse Operations

Inverse operations are pairs of operations that reverse each other's effects. In the context of exponential equations, logarithms are the inverse operation of exponentiation.
When you have an exponential equation like \( b^x = y \), you can convert it into its logarithmic form \( \log_b(y) = x \) using inverse operation principles. This is because logarithms undo what exponentiation does, helping you to determine the unknown exponent.

• Understanding the "inverse" concept is about realizing that these operations are like undoing a function.
• If exponentiation is considered as building up a result through repeated multiplication, logarithms effectively break it down to find the power or exponent involved.

Using inverse operations makes complex equations simpler to solve. So, whenever you need to solve for the exponent, remember that you'll likely be taking the logarithm of the result.

Base and Exponent Concept

In any exponential equation \( b^x = y \), understanding the roles of the base and the exponent is critical. The base \( b \) is the starting number or factor, and it is consistent within the equation. The exponent \( x \) signifies how many times the base will be multiplied by itself to reach the result \( y \).
This concept is fundamental because it underpins the entire structure of how exponential equations operate.

• The base can be any positive number, but it must remain constant across any comparisons or calculations within the equation.
• Exponents must be carefully calculated as they determine the scale or magnitude of the result.

Certain bases like 10 or 2 are common due to their mathematical properties and interpretations in practical settings, like logarithmic scales and binary systems. In the given exercise, recognizing 5 as the base and 7 as the exponent allowed for easy translation into a logarithmic expression, showcasing their inherent interconnectedness.