Problem 150 - Precalculus

Question

For the following exercises, solve for \(x\) by converting the logarithmic equation to exponential form. $$\log _{5}(x)=2$$

Step-by-Step Solution

Answer

The solution is \( x = 25 \).

Step 1: Understand the Logarithmic Equation

The given equation is \( \log_{5}(x) = 2 \). This is a logarithm with base 5.

Step 2: Convert to Exponential Form

To convert the logarithmic form \( \log_{b}(a) = c \) to exponential form, use the formula \( b^{c} = a \).In this exercise, it becomes \( 5^{2} = x \).

Step 3: Calculate the Exponential Expression

Calculate \( 5^2 \). Thus, \( 5 \times 5 = 25 \).

Step 4: Obtain the Solution for x

From the exponential form, we have \( x = 25 \).

Key Concepts

Exponential Form

Exponential form is a way to express numbers and equations using powers or exponents. In the context of converting a logarithm to exponential form, the logarithm gives us a clear base, result, and exponent. A logarithmic equation like \( \log_{b}(a) = c \) tells us that \( b^c = a \), where:

• \( b \) is the base of the logarithm and also the base in the exponential expression.
• \( c \) is the exponent.
• \( a \) is the result of the exponential expression.

In our example, \( \log_{5}(x) = 2 \), we read this as "log base 5 of \( x \) equals 2." By transforming this into exponential form, we get \( 5^{2} = x \). This step is crucial because it simplifies solving for \( x \) by putting it into a form that deals with basic multiplication instead of logarithms.

Solving for x

Once the equation is in exponential form, solving for \( x \) becomes straightforward. Now the equation is \( 5^2 = x \). To find \( x \), we simply need to calculate \( 5^2 \). Here's how you do it:

• Multiply 5 by itself: \( 5 \times 5 = 25 \).

Now we can see clearly that \( x = 25 \). This method is effective because it leverages basic arithmetic to resolve what was initially a logarithmic equation. The ability to switch from logarithmic to exponential form means using familiar multiplication to find the unknown value.

Base of Logarithm

The base of a logarithm is an integral part of both logarithmic and exponential forms. It is the number that is raised to a power to yield a specific value. In the equation \( \log_{5}(x) = 2 \), the base is 5. Understanding the base is vital since it serves as the foundation for your calculations:

• In \( 5^{2} = x \), the number 5 is what is repeatedly multiplied by itself, 2 times, which results in the value of \( x \).
• This highlights how the base dictates the exponential growth or decline; larger bases grow larger much faster as the exponent increases.

By recognizing the base in the logarithmic equation, converting to exponential form becomes more intuitive, letting you generally understand how the components fit together.