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Chapter 4: Problem 16

solve for \(x\) without using a calculating utility. $$\log _{10}(1+x)=3$$

Short Answer

Expert verified
The solution is \( x = 999 \).

Step by step solution

01

Understand the problem

We need to solve the equation \( \log_{10}(1+x) = 3 \) for \( x \). This means we need to find the value of \( x \) that makes this equation true.
02

Convert from logarithmic to exponential form

The equation \( \log_{10}(1+x) = 3 \) can be rewritten in exponential form. Recall that \( \log_b(a) = c \) is equivalent to \( a = b^c \). Thus, \( 1 + x = 10^3 \).
03

Calculate the exponential

Calculate \( 10^3 \). Since \( 10^3 = 10 \times 10 \times 10 = 1000 \), we have \( 1 + x = 1000 \).
04

Solve for x

Subtract 1 from both sides of the equation \( 1 + x = 1000 \) to isolate \( x \). This gives \( x = 1000 - 1 \).
05

Simplify

Carry out the subtraction: \( 1000 - 1 = 999 \). Thus, \( x = 999 \).

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Key Concepts

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

Exponential Form
Understanding how to convert a logarithmic equation into its exponential form is essential. The logarithmic equation \( \log_{10}(1+x) = 3 \) needs to be rewritten using the basic relationship between logs and exponents.
  • A logarithm \( \log_b(a) = c \) signifies that \( b \) raised to the power of \( c \) gives \( a \).
  • Therefore, \( \log_{10}(1+x) = 3 \) becomes \( 1+x = 10^3 \).
Mapping the logarithmic expression to an exponential one is a fundamental technique to prepare the equation for further simplifications. This step allows us to solve the equation by dealing with simpler operations like multiplication.
Solving for x
Transforming equations to make \( x \) the subject is vital when asked to solve for \( x \). Once we have the exponential form of the equation, like \( 1 + x = 1000 \), we can isolate \( x \) through basic arithmetic operations.
  • Start by identifying the operation associated with \( x \). In our case, it is addition by 1.
  • To isolate \( x \), perform the inverse operation, which is subtraction, on both sides. We thus subtract 1 from 1000.
  • This yields \( x = 999 \).
It's important to always perform the same operation on both sides of an equation. This ensures that the equation stays balanced, ultimately leading to the solution.
Basic Logarithm Properties
Grasping basic logarithm properties allows you to work more effectively with these functions. Being familiar with how logs translate helps in many math problems.
  • A logarithm \( \log_b(a) = c \) represents the exponent to which the base \( b \) must be raised to obtain \( a \).
  • Consequently, \( 10^3 = 1000 \) exemplifies how exponentiation is the inverse operation of taking the logarithm.
  • This relationship allows for the switch from logarithmic form to exponential form, making equations easier to handle and solve.
Additionally, the base of the logarithm (in this case, 10) often simplifies manipulations because raising 10 to any power involves straightforward multiplication. Recognizing these patterns can save time and aid comprehension.

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Most popular questions from this chapter

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