Problem 64 - College Algebra and Calculus: An Applied Approach

Question

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\log _{10} x=-5\)

Step-by-Step Solution

Answer

The solution to the equation is \(x=10^{-5}\), which is approximately \(0.00001\).

Step 1: Write down the given equation

Rewrite the problem. The following is the logarithmic equation given in the problem: \( \log _{10} x=-5 \)

Step 2: Translate into an exponential form

Transform the logarithmic expression into an exponential expression. By definitions of logarithms, a base-10 log equation \(\log _{10} x=-5\) can be written in exponential form as \(10^{-5}=x\).

Step 3: Calculate the value of x

After converting the equation into exponential form, it's straightforward to solve for \(x\). Calculate \(10^{-5}\) to find the value of \(x\).

Key Concepts

Solving Logarithms

Understanding how to solve logarithmic equations is crucial for students as they underpin many concepts in mathematics and science. A logarithmic equation contains a logarithmic function that has an unknown variable, which you need to solve for. The general format of a logarithm is \( \log_b x = y \), which reads as 'log base \( b \) of \( x \) equals \( y \)'.

When solving a logarithmic equation like \( \log _{10} x=-5 \), your goal is to isolate the variable. This usually involves converting the logarithm into exponential form, which allows you to more easily see what \( x \) equals. To isolate \( x \) in our example, you rewrite the equation without the logarithm, which will be detailed in the next section.

Exponential Form Conversion

Logarithms and exponentials are inverses of each other, which is why converting between these two forms is a foundational skill. To convert a logarithmic equation into exponential form, you use the basic definition of a logarithm. The equation \( \log_b x = y \), when converted to exponential form, becomes \( b^y = x \).

Let's apply this to the given problem \( \log _{10} x=-5 \). Following the definition, we have the base \(10\)) and our \( y \) value is \( -5 \) from the logarithmic equation. So, in exponential form, our equation is \( 10^{-5} = x \), which means \( x \) is equal to \( 10 \) raised to the power of \( -5 \). This conversion is pivotal for solving the problem as it directly leads us to a numeric value for \( x \) in the next step.

Logarithmic to Exponential

Once you have the exponential form of the equation, the next step is to actually solve for the variable. In the context of our problem, once you've transformed \( \log _{10} x=-5 \) to \( 10^{-5} = x \), you need to evaluate \( 10^{-5} \). An exponential equation with a negative exponent indicates that the value is a fraction—in this case, \( 1 \over 10^5 \).

Therefore, \( x \) equals \( 0.00001 \), after evaluating \( 10^{-5} \). This relates back to the concept of logarithms being the inverse of exponentials because you've essentially 'unwound' the logarithm to find the original number that was raised to a power to reach the number within the log function.