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Chapter 11: Problem 9

Solve each equation. $$12+2 \ln (x)=14$$

Short Answer

Expert verified
x = e

Step by step solution

01

Isolate the logarithmic term

Start by isolating the term with the natural logarithm. Subtract 12 from both sides of the equation.\[12 + 2 \ln(x) = 14 \]Subtract 12:\[2 \ln(x) = 2\]
02

Divide by the coefficient of the logarithm

Divide both sides by 2 to solve for the natural logarithm term.\[\ln(x) = 1\]
03

Exponentiate both sides

Exponentiate both sides to eliminate the natural logarithm. Recall that if \(\ln(x) = y\), then \(x = e^y\).\[x = e^1\]Therefore:\[x = e\]

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Key Concepts

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

natural logarithms
Natural logarithms are a special kind of logarithms. They use the mathematical constant 'e' (approximately 2.718) as their base. We write the natural logarithm of a number, say x, as ln(x).

When you see ln(x), it means: 'To what power should 'e' be raised to get x?' In other words, if e^y = x, then ln(x) = y. This property makes natural logarithms very powerful tools in algebra and calculus.

Remembering the relationship between logarithms and exponentiation is essential when dealing with logarithmic equations.
exponentiation
Exponentiation is the process of raising a number to a power. For example, if we have x raised to the power of y, it's written as x^y.

In our example, after we've simplified the logarithmic equation to ln(x) = 1, we use exponentiation to solve for x.

Since ln(x) = y implies x = e^y, we can raise 'e' to the power of 1 (because ln(e) = 1), resulting in x = e. This step allows us to 'remove' the logarithm and solve for the variable directly.
isolating variables
Isolating the variable is a crucial step in algebra, especially in equations with logarithms. Here鈥檚 how it鈥檚 done in our example:

We start with the equation 12 + 2 ln(x) = 14. Our first goal is to get the ln(x) term alone on one side of the equation. We do this by subtracting 12 from both sides, giving us 2 ln(x) = 2.

Next, we divide both sides by 2. This step leaves us with ln(x) = 1. By isolating ln(x), we prepare ourselves to apply the exponential function effectively in the next step.
algebraic manipulation
Algebraic manipulation involves using a set of algebraic rules to transform and solve equations. When dealing with logarithmic equations:
- Move terms to isolate variables using addition or subtraction.
- In our example, subtract 12 from both sides: 12 + 2 ln(x) = 14 becomes 2 ln(x) = 2.

- Divide both sides by the coefficients of the logarithm term: Divide by 2 to get ln(x) = 1.

These steps are fundamental to manipulating equations, making it easier to switch forms and solve for the unknown variable.

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Most popular questions from this chapter

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$8^{x}=2^{x+1}$$

For each equation, find the exact solution and an approximate solution when appropriate. Round approximate answers to three decimal places. See the Strategy for Solving Exponential and Logarithmic $$2^{x}=\frac{1}{3}$$

Use a calculator to evaluate each logarithm. Round answers to four decimal places. $$\ln (6.238)$$

Solve each equation. $$e^{2 x}=7$$

Which of the following expressions is not equal to \(\log \left(5^{2 / 3}\right) ?\) Explain. a) \(\frac{2}{3} \log (5)\) b) \(\frac{\log (5)+\log (5)}{3}\) c) \((\log (5))^{2 / 3}\) d) \(\frac{1}{3} \log (25)\)
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