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Chapter 5: Problem 18

Solve for \(x\) or \(b\). (a) \(\log _{b} 27=3\) (b) \(\log _{b} 125=3\)

Short Answer

Expert verified
The solutions to the equations are \( b = 3 \) and \( b = 5 \) respectively.

Step by step solution

01

Define the Logarithm as an Exponent

In the first equation, \(\log _{b} 27=3\), the logarithm is equal to the exponent 3. So, according to the base exponent rule, this equation is equivalent to \(b^3 = 27\).
02

Solve for the Base of the Logarithm

Next, solve \(b^3 = 27\) by taking the cube root of both sides of the equation. This yields \(b = \sqrt[3]{27}\).
03

Calculate the Value of the Base

On simplifying \(\sqrt[3]{27}\), we get \(b = 3\).
04

Define the Second Logarithm as an Exponent

In the second equation, \(\log _{b} 125=3\), again the logarithm is equal to the exponent 3. So, this equation is equivalent to \(b^3 = 125\).
05

Solve for the Base of the Second Logarithm

Next, solve \(b^3 = 125\) by taking the cube root of both sides of the equation. This yields \(b= \sqrt[3]{125}\).
06

Calculate the Value of the Base

On simplifying \( \sqrt[3]{125} \), we get \( b = 5 \).

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Key Concepts

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

Cube Root
The cube root of a number is a special value which, when multiplied by itself twice, gives the original number. For example, the cube root of 27 is 3, because when you multiply 3 by itself twice (3 \(\times\) 3 \(\times\) 3), you get 27.

Here鈥檚 how to find a cube root:
  • Identify a number you want to find the cube root for (e.g., 27).
  • Think of a number that when raised to the power of 3 (cubed), equals your original number.
  • In this case, 3 cubed is 27, so the cube root of 27 is 3.
Cube roots are essentially the inverse operation of cubing, just like how a square root is the inverse of squaring a number. They are useful in solving equations where a value is raised to the power of three.
Exponentiation
Exponentiation is a mathematical operation involving a base and an exponent. The operation is written as \(b^n\), where \(b\) is the base and \(n\) is the exponent. This means multiplying the base by itself \(n\) times.

For example, \(b^3\) tells us to multiply \(b\) by itself twice (\(b \times b \times b\)). If \(b = 2\), \(b^3 = 2 \times 2 \times 2 = 8\).
  • When the exponent is positive, it tells how many times to use the base in multiplication.
  • When the exponent is zero, the result is 1, as any non-zero number to the power of zero is 1.
  • If the exponent is negative, it represents the reciprocal of the base raised to the positive exponent.
Understanding exponentiation is crucial for dealing with logarithms because logarithms are essentially the inverse of exponentiation. In solving logarithmic equations, rewriting them in the form of an exponent is a key step to finding the unknown value.
Solving Equations
Solving equations involves finding an unknown variable that satisfies the mathematical statement you are given. In the context of logarithms, solving equations often involves exponentiation and inverse operations.

A typical approach involves the following steps:
  • First, express the logarithmic equation using exponentiation. For instance, a logarithmic equation like \(\log_b x = y\) can be written in exponential form as \(b^y = x\).
  • Next, use algebraic manipulations, such as taking roots or factoring, to isolate the variable you are solving for. In our given exercises, this involved taking the cube root of both sides of the equation.
  • Finally, calculate the value of the unknown by simplifying the equation to its simplest form.
This process often requires understanding the properties of exponents and roots, especially when conversions between different mathematical representations are necessary. Mastery of these techniques is crucial for solving more complex equations that involve logarithmic and exponential functions.

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