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Chapter 4: Problem 49

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log (p+17)=4.1\)

Short Answer

Expert verified
The exact solution is \( p = 10^{4.1} - 17\). The approximate solution is \( p \approx 12572.2541\).

Step by step solution

01

- Isolate the logarithm

The given equation is \(\text{log} (p+17) = 4.1\). The logarithm is already isolated on one side of the equation, so we can proceed to the next step.
02

- Rewrite the equation in exponential form

To eliminate the logarithm, rewrite the equation in its exponential form. Since the base of \(\text{log} \) is 10, the equation becomes: \(\text{p} + 17 = 10^{4.1}\).
03

- Solve for p

To solve for \(p\), subtract 17 from both sides of the equation: \( p = 10^{4.1} - 17\).
04

- Calculate the exact and approximate solutions

First, find the exact value of \(10^{4.1}\), then subtract 17 to get the exact solution: \( p = 10^{4.1} - 17 \). Next, calculate the approximate value of \(10^{4.1}\) which is approximately 12589.2541. Thus, \( p = 12589.2541 - 17 \approx 12572.2541 \).

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Key Concepts

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

Exponential Form
When we have a logarithmic equation like \(\text{log} (p+17) = 4.1\), we need to convert it to its exponential form to solve for the variable. In general, if you have an equation in the form \(log_b(x) = y\), it can be rewritten in the exponential form as \(b^y = x\). Here, the base is 10 (since the log with no base specified is a common logarithm), so \(\text{log} (p+17) = 4.1\) transforms into \({10}^{4.1} = p+17\). By rewriting in this form, we eliminate the logarithm and make the problem easier to solve.
Logarithmic Isolation
To solve a logarithmic equation, first isolate the log term. In our example, the logarithm is already isolated: \(log (p+17) = 4.1\). Isolation means having the log expression on one side alone. If it wasn't isolated, you would need to move other terms to the opposite side of the equation. This could involve steps such as adding, subtracting, multiplying, or dividing other terms.
Approximate Solutions
Sometimes, an exact solution can be cumbersome or nearly impossible to use practically, so we find an approximate solution. After transforming and isolating terms, we calculate the exact solution as \({10}^{4.1} - 17\). We then use a calculator to find \({10}^{4.1}\), which is approximately 12589.2541. Subtract 17 to get an approximate value of 12572.2541. It's essential to round it correctly, as specified. Here, rounding to four decimal places gives us \(p \approx 12572.2541\). So, whenever you're given the task to find approximate solutions, make sure to follow the rounding rules carefully.

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