Question

In Problems \(17-24\), solve for \(x .\) Hint: \(\log _{a} b=c \Leftrightarrow a^{c}=b\). \(\log _{2} 8=x\)

Short answer

The solution is \(x = 3\).

Step-by-step solution

Identify the logarithmic equation

The given equation is \( \log_{2} 8 = x \). Here, the logarithmic form provides us the base \(a = 2\), and the argument \(b = 8\). Our task is to find the exponent \(x\) such that \(2^x = 8\).

Rewrite using the definition of logarithms

Using the hint provided, we can rewrite the equation \( \log_{a} b = c \) as \( a^{c} = b \). In our specific case, this becomes \( 2^x = 8 \).

Express the right side in the same base

Recognize that \(8\) can be expressed as a power of \(2\). We know \(8 = 2^3\). Therefore, the equation \(2^x = 8\) can be rewritten as \(2^x = 2^3\).

Solve the equation by comparing exponents

Since the bases are the same, we can equate the exponents: \(x = 3\).

Key concepts

Exponents

Exponents are a way to express repeated multiplication of a number by itself. If you see a number expressed as a base with a small number in a superscript, that small number is the exponent. For example, in the expression \(2^3\), 2 is the base and 3 is the exponent, meaning we multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).
Understanding exponents is crucial because they are used in various mathematical computations, allowing us to simplify complex multiplications into an easier-to-read format.

• **Base:** The number being multiplied repeatedly.
• **Exponent:** Tells us how many times the base is used as a factor.
• **Exponential form:** The compact way of expressing repeated multiplication (e.g., \(2^3\)).

Knowing these fundamentals helps when you encounter problems involving powers, such as rewriting numbers to the same base, or finding unknown exponents when they relate to logarithmic problems.

Logarithms

Logarithms can be thought of as the opposite of exponents. They answer the question: How many times must we multiply a given number (base) to get another number? For instance, in the expression \( \log_{2} 8 \), we're seeking the exponent to which 2 must be raised to result in 8.
The general form of a logarithmic equation is \(\log_{a} b = c\), which transforms into \(a^c = b\). In this setup:

• **Logarithm:** The process of finding the exponent (\(c\)) that relates a base (\(a\)) to its power (\(b\)).
• **Base (\(a\)):** The number you're raising to the power.
• **Argument (\(b\)):** The result that comes from the base raised to the exponent.

Logarithms are helpful in many fields such as science and engineering, because they simplify the handling and interpretation of large numbers, and make computations easier when dealing with exponential growth or decay.

Equation Solving

Equation solving, especially when involving exponents and logarithms, combines several arithmetic skills. The goal is to find the unknown value that makes the equation true. In our example, we convert the logarithmic equation \( \log_{2} 8 = x \) into an exponential equation: \(2^x = 8\).
Here's how you can effectively solve such equations:

• **Identify:** Recognize the form of the given equation. Is it a logarithmic or exponential equation?
• **Transform:** Use logarithmic identities or definitions to rewrite the equation. For example, convert \(\log_{a} b = c\) to \(a^c = b\).
• **Simplify:** If possible, express both sides with the same base. This aids in direct comparison.
• **Solve:** Equate the exponents (if the bases are the same) or perform operations to isolate the variable.

This structured approach empowers you to tackle various logarithmic and exponential equations with confidence, ensuring you can find the variable in any algebraic context.