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Chapter 0: Problem 16

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

Short Answer

Expert verified
The value of \( x \) is 999.

Step by step solution

01

Apply the Definition of Logarithms

The logarithmic equation \( \log_{10}(1+x)=3 \) can be rewritten using the definition of logarithms. The definition states that if \( \log_b(a) = c \), then \( b^c = a \). Here, we have \( b = 10 \), \( a = 1+x \), and \( c = 3 \). Thus, the equation becomes \( 10^3 = 1+x \).
02

Solve for x

Now we need to solve the equation \( 10^3 = 1+x \). We know that \( 10^3 = 1000 \), so we substitute this into the equation: \( 1000 = 1 + x \).
03

Isolate x

To find the value of \( x \), subtract 1 from both sides of the equation: \( x = 1000 - 1 \).
04

Compute the Final Result

Perform the subtraction: \( x = 999 \).

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Key Concepts

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

Definition of Logarithms
Understanding the definition of logarithms is crucial when dealing with problems that involve them. A logarithm answers the question: "To what power must the base be raised to produce a given number?" More mathematically, given a logarithmic equation of the form \(\log_b(a) = c\), the base \(b\) raised to the power \(c\) gives the number \(a\). This can be rewritten in exponential form as \(b^c = a\). For example, in the equation \( \log_{10}(1+x) = 3\), the base is 10, the number is \(1+x\), and it equals 3. Hence, we express it as \(10^3 = 1+x\). This swap from a logarithmic form to an exponential form often makes it easier to solve the equation.
Solving Logarithmic Equations
After converting a logarithmic equation into an exponential form using the definition, the next step is solving it. In the current example, after we convert \(\log_{10}(1+x) = 3\) to \(10^3 = 1+x\), we solve for \(x\). The key to solving logarithmic equations is shifting from a logarithmic to an arithmetic mindset: turn the complex looking log equation into something more familiar and easier to solve.

In our example, \(10^3\) calculates to 1000. This means the equation simplifies to \(1000 = 1+x\). The next logical step is to isolate \(x\), which involves straightforward algebraic manipulation.
Mathematical Operations
Algebraic manipulation is at the heart of solving equations. Once we have the equation in the form \(1000 = 1+x\), the next step is to isolate the variable \(x\). To do this, one can simply perform the operation of subtraction: subtract 1 from both sides of the equation.

This gives us:
  • \(1000 - 1 = x\)
  • Thus: \(x = 999\)
Mathematical operations such as addition, subtraction, multiplication, and division are basic tools that allow us to manipulate equations and solve for unknowns. Understanding and applying these operations correctly is essential in reaching the solution.

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Most popular questions from this chapter

Determine whether the statement is true or false. Explain your answer. The natural logarithm function is the logarithmic function with base \(e\).

On the Richter scale, the magnitude \(M\) of an earthquake is related to the released energy \(E\) in joules (J) by the equation $$ \log E=4.4+1.5 M $$ (a) Find the energy \(E\) of the 1906 San Francisco earthquake that registered \(M=8.2\) on the Richter scale. (b) If the released energy of one earthquake is 10 times that of another, how much greater is its magnitude on the Richter scale?

Suppose that the function \(f\) has domain all real numbers. Determine whether each function can be classified as even or odd. Explain. (a) \(g(x)=\frac{f(x)+f(-x)}{2}\) (b) \(h(x)=\frac{f(x)-f(-x)}{2}\)

Solve for \(x\) without using a calculating utility. Use the natural logarithm anywhere that logarithms are needed. $$ 2 e^{3 x}=7 $$

In each part, identify the domain and range of the function, and then sketch the graph of the function without using a graphing utility. (a) \(f(x)=1-e^{-x+1}\) (b) \(g(x)=3 \ln \sqrt[3]{x-1}\)
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