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Chapter 9: Problem 23

Find the exact value. See Examples 2 and \(6 .\). \(\log 10^{3}\)

Short Answer

Expert verified
The exact value is 3.

Step by step solution

01

Understand the Logarithmic Function

The logarithmic function \(\log_b(a)\) finds the exponent \(x\) such that the base \(b\) raised to \(x\) equals \(a\). For this exercise, we have \(\log_{10}(10^3)\), which asks what power of 10 gives \(10^3\).
02

Apply the Power Rule of Logarithms

The power rule of logarithms states that \(\log_b(a^c) = c \cdot \log_b(a)\). Applying this to \(\log_{10}(10^3)\), we get \(3 \cdot \log_{10}(10)\).
03

Evaluate the Basic Logarithm

We know that \(\log_{10}(10) = 1\) because 10 raised to the power of 1 equals 10. Substituting this value back, we have \(3 \cdot 1\).
04

Calculate the Expression

Multiply the numbers from the previous step to find the exact value: \(3 \cdot 1 = 3\). Thus, the exact value of \(\log_{10}(10^3)\) is 3.

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Key Concepts

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

Power Rule of Logarithms
The power rule of logarithms is a super handy trick. It helps simplify problems involving logarithms raised to powers. Here鈥檚 what it says: if you take a logarithm of something in the form of \(a^c\), you can rewrite it as \(c \cdot \log_b(a)\). This is powerful because it turns complex power expressions into simple multiplication.
For example, take the expression \(\log_{10}(10^3)\). Using the power rule, you can move the exponent 3 in front of the logarithm. So it becomes \(3 \cdot \log_{10}(10)\). This simplifies things instantly.
  • Break down larger numbers into bases you can easily handle.
  • Use this rule to turn exponentials into products.
This rule is brilliant for tidy calculations, especially in scientific and engineering fields where it鈥檚 all about simplifying math.
Exponents
Exponents are a way to express repeated multiplication of the same number by itself. They are seen as a small number placed to the upper right of a base number. For example, \(10^3\) means multiplying 10 by itself twice, which results in 1000.
Exponents follow some basic rules:
  • Multiplication Rule: When multiplying numbers with the same base, add the exponents: \(a^m \cdot a^n = a^{m+n}\).
  • Division Rule: When dividing numbers with the same base, subtract the exponents: \((a^m)/(a^n) = a^{m-n}\).
  • Power of a Power Rule: When raising a power to another power, multiply the exponents: \( (a^m)^n = a^{m \cdot n}\).
Understanding these rules of exponents helps in simplifying expressions and solving equations efficiently.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions. They are tremendously useful in solving equations where the variable is an exponent. The basic form is \(\log_b(a) = x\), which reads as "the base \(b\) raised to the power of \(x\) equals \(a\)."
Logarithmic functions have their own set of properties:
  • Product Rule: \(\log_b(xy) = \log_b(x) + \log_b(y)\).
  • Quotient Rule: \(\log_b(x/y) = \log_b(x) - \log_b(y)\).
  • Change of Base Formula: \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), useful for calculating logs with a new base.
Logarithmic functions often appear in diverse scientific concepts, from measuring sound intensity to exponential decay. They make complex multiplication and division more manageable by converting them into simpler addition and subtraction tasks.

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Most popular questions from this chapter

The formula \(\log _{10}(1-k)=\frac{-0.3}{H}\) models the relationship between the half-life \(H\) of a radioactive material and its rate of decay \(k .\) Find the rate of decay of the iodine isotope \(\mathrm{I}-131\) if its half-life is 8 days. Round to four decimal places.

Simplify each rational expression. See Section 6.1. $$ \frac{x^{2}-8 x+16}{2 x-8} $$

When solving a logarithmic equation, explain why you must check possible solutions in the original equation.

CONCEPT EXTENSIONS. Use a calculator to try to approximate log 0. Describe what happens and explain why.

Graph each function by finding ordered pair solutions, plotting the solutions, and then drawing a smooth curve through the plotted points. $$ f(x)=\log (x-2) $$
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