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Chapter 5: Problem 14

Express each exponential equation as a logarithmic equation. $$10^{\circ}=1$$

Short Answer

Expert verified
The equation is \( \log_{10}(1) = 0 \).

Step by step solution

01

Identify the form of the exponential equation

The given equation is in the form \( a^{b} = c \), where \( a = 10 \), \( b = 0 \), and \( c = 1 \). We need to rewrite this equation in logarithmic form.
02

Rewrite in logarithmic form

The logarithmic form of an equation \( a^{b} = c \) is \( \log_{a}(c) = b \). Replace \( a, b, c \) with the values from the given equation.
03

Substitute values into logarithmic form

Substitute \( a = 10 \), \( b = 0 \), and \( c = 1 \) into the logarithmic form equation to get \( \log_{10}(1) = 0 \).

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Key Concepts

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

Exponential Equations
Exponential equations are fundamental in math and science. These types of equations have the form \( a^b = c \), where \( a \) is the base, \( b \) is the exponent, and \( c \) is the result or outcome. In simpler terms, an exponential equation shows how many times we multiply a number, the base, by itself to get another number.Here are a few important points about exponential equations:
  • The base \( a \) is typically a positive number.
  • The exponent \( b \) can be a positive, negative, or even a fraction.
  • The result \( c \) is the outcome of raising the base to the power of the exponent.
To solve exponential equations, we often need to convert them to logarithmic form, because logarithms "undo" exponentiation, much like subtraction "undoes" addition. Understanding how to transition between these two forms is crucial for solving many mathematical problems.
Logarithms
Logarithms are the inverse operations of exponentiation. Just as subtraction is the inverse of addition, logarithms allow us to "work backwards" from the result of an exponentiation to find the missing exponent. When dealing with logarithms and exponentials, it's crucial to remember:
  • \( \log_a(c) = b \) means the same as \( a^b = c \). This relationship is key.
  • The base of the logarithm \( a \) must be positive and not equal to one, as it represents how many times to multiply to reach \( c \).
Logarithms come in handy for solving equations where the unknown is an exponent. They simplify problems by transforming exponential growth patterns into linear or manageable equations. By learning logarithms, you acquire a significant tool, especially since these concepts apply widely in science, finance, and technology sectors.
Base 10 Logarithm
Base 10 logarithms, also known as common logarithms, are a prevalent type of logarithms, especially useful in everyday applications. Denoted as \( \log_{10}(x) \), common logs quantify how many times one must multiply 10 to achieve the number \( x \).Key attributes of base 10 logarithms include:
  • They are convenient for calculations with numbers that are multiples or fractions of 10.
  • \( \log_{10}(10) = 1 \) because 10 raised to the power of 1 equals 10.
  • \( \log_{10}(1) = 0 \) because any number to the power of 0 is 1.
Base 10 logarithms are particularly popular for creating scales and graphs, like the Richter scale for earthquakes or decibel levels for sound. Furthermore, they simplify complex multiplication into basic addition, a key property extensively utilized in scientific calculators and computational software.

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Most popular questions from this chapter

Use the change of base formula and common or natural logarithms to evaluate each logarithmic expression. Approximate your answer to 3 significant figures. $$\log _{\frac{1}{2}} 40$$

Use a GDC to evaluate the expression, correct to 4 significant figures. $$\log 25$$

Find the exact value of the expression without using your GDC. $$\log _{8} 1$$

Write each expression as the logarithm of a single quantity. $$\log x-\log y-\log z$$

Find the exact value of the expression without using your GDC. $$\ln (\sqrt{e})$$
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