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Chapter 5: Problem 27

Solve each equation. Give the exact answer. $$\log _{2}(x+1)=3$$

Short Answer

Expert verified
The solution is x = 7.

Step by step solution

01

Understand the Logarithmic Equation

The equation given is \( \log_{2}(x+1) = 3 \). This is a logarithmic equation with base 2, which suggests 'if 2 is raised to the power of 3, the result should be \( x+1 \)'.
02

Use the Definition of Logarithms

By the definition of logarithms, \( \log_{b}(a) = c \) implies \( b^{c} = a \). Apply this to the equation \( \log_{2}(x+1) = 3 \), which gives \( 2^{3} = x+1 \).
03

Calculate the Exponent

Calculate \( 2^{3} \), which equals 8. So, the equation simplifies to \( 8 = x + 1 \).
04

Solve for x

Rearrange the equation \( 8 = x + 1 \) to solve for \( x \). Subtract 1 from both sides to find \( x = 7 \).
05

Verify the Solution

Plug \( x = 7 \) back into the original equation: \( \log_{2}(7+1) = \log_{2}(8) \). Since \( \log_{2}(8) = 3 \) (because \( 2^{3} = 8 \)), the solution is correct.

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Key Concepts

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

Definition of Logarithms
A logarithm answers the question: "To what power must the base be raised, to achieve a certain number?" In general terms, this is expressed as \( \log_{b}(a) = c \), where \( b \), the base, if raised to an exponent \( c \), will yield \( a \).
For example, in the equation \( \log_{2}(8) = 3 \), the base 2 must be raised to the power 3 to get 8. Therefore, logarithms are a way to express exponents in another form.

Some key points about logarithms include:
  • The base of a logarithm must always be greater than zero and cannot be equal to one.
  • The argument of the logarithm (in this case \( a \)) must be positive.
  • Logarithms are often used to solve exponential equations by converting them into a more manageable form.
Understanding how logarithms work by their very definition is crucial in algebra and beyond. It sets the foundation for transforming complex equations into simpler, solvable forms.
Solving Equations
When it comes to solving logarithmic equations, like the one in your exercise \( \log_{2}(x+1) = 3 \), the first step is to interpret the logarithmic expression using its definition. This involves rewriting it as an exponent.
This equation translates to \( 2^{3} = x+1 \), indicating you need \( x+1 \) to equal the result of \( 2^{3} \).

To solve this:
  • First, compute the exponent: Calculate \( 2^{3} \), which gives 8.
  • Next, set the expression equal to 8: This means \( x + 1 = 8 \).
  • Finally, solve for \( x \): Subtract 1 from both sides to isolate \( x \), resulting in \( x = 7 \).
The goal is to isolate \( x \) through these straightforward operations. Checking your solution by substituting back into the original equation confirms its accuracy, ensuring that no steps were overlooked.
Exponents
Exponents are a way to express repeated multiplication of a base number. For example, \( 2^{3} \) means multiplying 2 by itself three times: \( 2 \times 2 \times 2 \), which equals 8.
In terms of logarithms, the exponent is what the logarithm essentially solves for. This concept is key in converting logarithmic expressions to their equivalent exponential form and vice versa.

Some important properties of exponents include:
  • The product of powers rule: \( b^{m} \times b^{n} = b^{m+n} \).
  • The power of a power rule: \( (b^{m})^{n} = b^{m \times n} \).
  • The power of a product rule: \( (ab)^{n} = a^{n} \times b^{n} \).

Understanding these properties helps simplify and solve both logarithmic and exponential equations. They allow you to rewrite expressions in ways that can be more easily managed, crucial for algebraic problem-solving.

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Most popular questions from this chapter

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