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Chapter 12: Problem 66

Solve each equation. \(\log _{1 / 2}(2 x-1)=3\)

Short Answer

Expert verified
x = \(\frac{9}{16}\)

Step by step solution

01

Understand the logarithmic equation

The given equation is \(\text{log}_{1/2}(2x-1) = 3\). Recall that \(\text{log}_b(a) = c\) means \(b^c = a\).
02

Rewrite the equation in exponential form

Rewrite the equation \(\text{log}_{1/2}(2x-1) = 3\) using the exponential form: \(\left(\frac{1}{2}\right)^3 = 2x - 1\).
03

Simplify the exponential

Calculate \((1/2)^3\): \((1/2)^3 = 1/8\). So the equation becomes: \(\frac{1}{8} = 2x - 1\).
04

Solve for the variable

Add 1 to both sides of the equation: \(\frac{1}{8} + 1 = 2x\). Convert 1 to a fraction: \(\frac{1}{8} + \frac{8}{8} = \frac{9}{8}\). The equation is now \(\frac{9}{8} = 2x\).
05

Isolate the variable

Divide both sides by 2 to isolate x: \(\frac{9}{8} \/ 2 = x\). Simplify the right side to get \(x = \frac{9}{16}\).

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Key Concepts

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

logarithmic functions
Logarithmic functions are very important in mathematics. They help us solve equations where the variable is inside a logarithm. A logarithmic function is the inverse of an exponential function. The general form of a logarithmic function is \(\text{log}_b(a) = c\), which means that the base b raised to the power c equals a, or \[b^c = a\].
This connection between logarithms and exponents is key in solving logarithmic equations.
When you see \(\text{log}_{1/2}(2x - 1) = 3\), it tells you that \(1/2\) is the base, and \((2x - 1)\) is the number. The result, 3, is the exponent.
exponential form conversion
To solve logarithmic equations, we often convert them from logarithmic form to exponential form. This conversion is very straightforward if you remember that \(\text{log}_b(a) = c\) corresponds to \[b^c = a\].
For example, with the equation \(\text{log}_{1/2}(2x-1) = 3\), we convert it by saying, \(1/2\) to the power of 3 equals \(2x - 1\).
This gives us: \(\frac{1}{2}^3 = 2x - 1\). This step-by-step approach simplifies the problem and makes it easier to work with the equation.
isolating variables
Isolating the variable is a crucial part of solving equations. Once the equation is in exponential form, you need to isolate the variable to find its value.
For instance, from \(\frac{1}{8} = 2x - 1\), we add 1 to both sides to get \(2x = \frac{9}{8}\).
Finally, we divide both sides by 2. This gives \[x = \frac{9}{16}\].
This process of isolating the variable involves simple algebraic operations such as addition, subtraction, multiplication, and division.

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Most popular questions from this chapter

Solve each equation. \(\log _{3}(2 x+7)=4\)

Solve each equation. \(\log _{x} 1=0\)

Solve each equation. \(\log _{5} x=-3\)

Each of the following functions is one-to-one. Graph the function as a solid line (or curve), and then graph its inverse on the same set of axes as a dashed line (or curve). Complete any tables to help graph the functions. $$ f(x)=2 x+3 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. $$\log _{10} 2^{19}$$
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