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Chapter 11: Problem 11

Solve each equation. $$\log (x)+\log (5)=1$$

Short Answer

Expert verified
x = 2

Step by step solution

01

Combine Logarithms

Use the product rule of logarithms: \ \( \log a + \log b = \log(ab) \). Combine the logarithms on the left side of the equation: \ \( \log(x) + \log(5) = \log(5x) \). So the equation becomes: \ \( \log(5x) = 1 \).
02

Exponentiate Both Sides

To solve for \(x\), rewrite the equation in exponential form. The base of the logarithm is 10, so: \ \( 10^{\log(5x)} = 10^1 \). This simplifies to: \ \( 5x = 10 \).
03

Solve for x

Isolate \(x\) by dividing both sides of the equation by 5: \ \( x = \frac{10}{5} \). Simplify the fraction to find: \ \( x = 2 \).

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Key Concepts

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

product rule of logarithms
When dealing with logarithms, there's a very helpful rule called the product rule. This rule states that the sum of two logarithms with the same base can be combined into a single logarithm. The formula looks like this: \( \log a + \log b = \log(ab) \). Essentially, instead of adding two separate logs, you can multiply their arguments and take the log of the product.

In our example, we start with \( \log(x) + \log(5) = 1 \). By applying the product rule, we combine the left-hand side into one logarithm: \( \log(5x) \). Now the equation is simplified to: \( \log(5x) = 1 \). This makes it much easier to work with and is a crucial first step in solving the equation.
exponential form
Once you have combined the logarithms using the product rule, the next step is to convert the logarithmic equation into its exponential form. This is often the easiest way to isolate the variable you need to solve for. Logarithms and exponents are closely related; in fact, a logarithm is simply the inverse of an exponent.

For the equation \( \log(5x) = 1 \), we need to remember that the base of common logarithms (logarithms with no base written explicitly) is 10. So, our equation becomes: \( 10^{\log(5x)} = 10^1 \). By converting to this exponential form, you effectively 'undo' the logarithm, simplifying the equation to: \( 5x = 10 \). Now you're ready for the final step.
solving for x
After converting the logarithmic equation to exponential form, the final step is simple algebra to solve for the variable you're interested in. For the equation \( 5x = 10 \), you just need to isolate \(x\) on one side of the equation.

To do this, divide both sides of the equation by 5: \( x = \frac{10}{5} \). Simplify the right-hand side to find the value of \(x\): \( x = 2 \).

You've now solved for \(x\).

Remember, the key steps were: 1) Apply the product rule of logarithms, 2) Convert to exponential form, and 3) Solve for \(x\). Practice these steps, and solving logarithmic equations will become much easier.

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Most popular questions from this chapter

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