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Chapter 4: Problem 46

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(5 \log _{6}(7 w+1)=10\)

Short Answer

Expert verified
The solution is \( w = 5 \).

Step by step solution

01

- Isolate the logarithmic expression

Divide both sides of the equation by 5 to isolate the logarithmic expression: \( 5 \log_{6}(7w + 1) = 10 \) \( \log_{6}(7w + 1) = \frac{10}{5} = 2 \)
02

- Convert the logarithmic equation to exponential form

Rewrite the logarithmic equation \( \log_{6}(7w + 1) = 2 \) in exponential form: \( 7w + 1 = 6^2 \)
03

- Solve for the variable

Simplify the exponential expression and solve for \( w \): \( 7w + 1 = 36 \) \( 7w = 36 - 1 \) \( 7w = 35 \) \( w = \frac{35}{7} = 5 \)
04

- State the solution set

The exact solution is: \( w = 5 \) Since we have an exact solution, there's no need for an approximate solution with decimal places.

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Key Concepts

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

Understanding Logarithms
A logarithm is essentially the reverse operation of exponentiation. If you have a number, its logarithm gives you the power to which a specific base must be raised to get that number. For example, in the expression \(\text{log}_{b}(x) = y\), \(b^y = x\). Here, \(\text{log}_{b}(x)\) means 'the power to which we need to raise \(b\) to get \(x\)'. In the provided exercise, we deal with \(\text{log}_{6}(7w+1)\), meaning we are looking for the power to which 6 must be raised to obtain \(7w + 1\).
What is Exponential Form?
Converting logarithmic expressions to exponential form is a common technique used to simplify equations. In exponential form, you move from \( \text{log}_{b}(x) = y \) to \( b^y = x \). This step is crucial since it turns a seemingly complex logarithmic equation into a more familiar exponential equation. In our exercise, we converted \( \text{log}_{6}(7w + 1) = 2 \) into its exponential form \( 7w + 1 = 6^2 \). As a result, we simplify our work by handling powers and basic arithmetic instead of navigating logarithms.
Isolating Variables
Isolating the variable is one of the most essential steps in solving any equation. To isolate the variable means to get the variable by itself on one side of the equation. This often requires performing inverse operations. In the given exercise, we start by isolating the logarithmic term \( \text{log}_{6}(7w + 1) \) by dividing both sides of the original equation by 5. This changes our equation from \( 5 \text{log}_{6}(7w + 1) = 10 \) to \( \text{log}_{6}(7w + 1) = 2 \). This simplification makes subsequent steps more straightforward.
Finding Exact Solutions
Exact solutions are critical for validating your understanding of the problem. It means finding the precise value without any approximations. Let's revisit the example: After converting to exponential form \( 7w + 1 = 36 \), we solve for \( w \) by isolating it: \( 7w = 36 - 1 \), then \( 7w = 35 \), and finally \( w = \frac{35}{7} = 5 \). This precise value confirms our exact solution. Once we have this, we don't need approximate values with decimal places, as requested in the exercise instructions.}

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Most popular questions from this chapter

Compare the graphs of \(Y_{1}=\frac{e^{x}-e^{-x}}{2}\), \(\mathrm{Y}_{2}=\ln \left(x+\sqrt{x^{2}+1}\right)\), and \(\mathrm{Y}_{3}=x\) on the viewing window [-15.1,15.1,1] by \([-10,10,1] .\) Based on the graphs, how do you suspect that the functions are related?

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(3^{6 x+5}=5^{2 x}\)

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{5} 3 $$

Determine if the statement is true or false. For each false statement, provide a counterexample. For example, \(\log (x+y) \neq \log x+\log y\) because \(\log (2+8) \neq \log 2+\log 8\) (the left side is 1 and the right side is approximately 1.204 ). $$ \log _{2}(7 y)+\log _{2} 1=\log _{2}(7 y) $$

Suppose that \(P\) dollars in principal is invested in an account earning \(3.2 \%\) interest compounded continuously. At the end of 3 yr, the amount in the account has earned \(\$ 806.07\) in interest. a. Find the original principal. Round to the nearest dollar. (Hint: Use the model \(A=P e^{r t}\) and substitute \(P+806.07\) for \(A .)\) b. Using the original principal from part (a) and the model \(A=P e^{r t},\) determine the time required for the investment to reach \(\$ 10,000\).
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