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Chapter 13: Problem 14

Express the given equations in logarithmic form. $$(81)^{3 / 4}=27$$

Short Answer

Expert verified
The equation in logarithmic form is \(\log_{81} 27 = \frac{3}{4}\).

Step by step solution

01

Identify the base, exponent, and result

In the equation \((81)^{3 / 4} = 27\), the base is 81, the exponent is \(\frac{3}{4}\), and the result is 27. These components will be used to form the logarithmic equation.
02

Apply logarithmic form

To convert the expression \((81)^{3 / 4} = 27\) into logarithmic form, use the relation: if \(a^b = c\) then it can be rewritten as \(\log_a c = b\).
03

Write the logarithmic equation

Substitute the identified components into the logarithmic form. The base \(a\) is 81, \(c\) is 27, and \(b\) is \(\frac{3}{4}\). Thus, the logarithmic form of the equation is \(\log_{81} 27 = \frac{3}{4}\).

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Key Concepts

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

Exponents
Exponents are a way to represent repeated multiplication of a number by itself. For instance, the expression \(a^n\) means that the number \(a\) is multiplied by itself \(n\) times. Here, \(a\) is called the base, and \(n\) is called the exponent. Exponents follow several important rules that can simplify calculations:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\). This rule says that when you multiply two numbers with the same base, you can add their exponents.
  • Power of a Power: \((a^m)^n = a^{m \cdot n}\). When raising an exponential number to another power, multiply the exponents.
  • Zero Exponent Rule: \(a^0 = 1\) for any \(a eq 0\). Any number raised to the power of zero equals one.
  • Negative Exponent Rule: \(a^{-n} = \frac{1}{a^n}\). A negative exponent indicates division.
In the equation \((81)^{3/4} = 27\), the base is 81, and the exponent is \(\frac{3}{4}\). This means that 81 is taken to a power of three-fourths, involving both a cube root and a square root.
Logarithms
Logarithms can be viewed as the inverse operation of exponents. They answer the question: "To what power must the base be raised, to produce a given number?" Mathematically, if \(a^b = c\), then \(b = \log_a c\). Here's how logarithms work:
  • The number we are raising is the base. In \( \log_{81} 27 \), 81 is the base.
  • In the expression, the number \(27\) is what we get after raising the base to a certain power. This is called the argument.
  • The result \(\frac{3}{4}\) is the logarithm of 27 with base 81, which represents the power to which the base must be raised.
Using logarithmic equations, you can easily solve for unknowns in exponential equations. For the given problem, converting \((81)^{3 / 4} = 27\) into logarithmic form as \(\log_{81} 27 = \frac{3}{4}\) helps us see the relationship clearly.
Mathematical Expressions
Mathematical expressions involve numbers, operations, and often variables combined to represent a calculation or value. Understanding expressions is crucial as they form the foundation of solving equations.
  • Components: An expression can include constants (like 81 or 27), variables, and operators (such as addition, multiplication).
  • Simplification: Expressions can often be rewritten in simpler or more informative forms, like converting an exponentiation problem into a logarithmic form for easier understanding.
  • Evaluation: Once an expression is simplified, it can be evaluated by substituting any known values for variables.
For the problem \((81)^{3/4} = 27\), we simplify by recognizing it as an exponential expression. Writing it in logarithmic form allows for easier interpretation and solution of complex equations.

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