Problem 104 - Precalculus

Question

Explain the mistake that is made. Evaluate the logarithm \(\log _{100} 10\) Solution: Set the logarithm equal to \(x\). \(\log _{100} 10=x\) Express the equation in exponential form. \(10^{x}=100\) Solve for \(x\). \(x=2\) Answer: \(\quad log_{100} 10=2\) This is incorrect. The correct answer is \(\log _{100} 10=\frac{1}{2}\) What went wrong?

Step-by-Step Solution

Answer

The mistake was treating \( 100^x = 10 \) incorrectly as \( 10^x = 100 \) instead of solving \( (10^2)^x = 10 \).

Step 1: Understand the Problem

We need to evaluate the logarithm \( \log_{100} 10 \). The expression means 'what power must 100 be raised to, in order to get 10?'.

Step 2: Set up the Logarithmic Equation

Set \( \log_{100} 10 = x \). This means \( 100^x = 10 \). Our task is to solve for \( x \).

Step 3: Express in Exponential Form

The given solution attempted to express \( 100^x = 10 \) as \( 10^x = 100 \), which is incorrect. The correct exponential form is \( 100^x = 10 \).

Step 4: Solve the Correct Exponential Equation

Rewrite 100 as a power of 10: \( 100 = 10^2 \). This gives us \( (10^2)^x = 10 \), which simplifies to \( 10^{2x} = 10^1 \).

Step 5: Equate the Exponents

Since the bases are equal, equate the exponents: \( 2x = 1 \).

Step 6: Solve for \( x \)

Divide both sides by 2 to solve for \( x \): \( x = \frac{1}{2} \).

Key Concepts

Exponential Equations

An exponential equation is one where a variable is in the exponent. These types of equations can seem tricky because we're used to seeing the variable as a base or a coefficient. The key to solving an exponential equation is to express each side with the same base, so you can then equate the exponents. This is because if two exponential expressions are equal and their bases are the same, then their exponents must also be equal. This concept was crucial in solving \/( \log_{100} 10 \/) where the mistake was converting the expression incorrectly.When solving an exponential equation:

• Try to express both sides of the equation using the same base.
• Once the bases are the same, set the exponents equal to each other.
• Solve the resulting simpler equation for the variable.

Remember, rewriting numbers with their most basic base (like expressing 100 as \(10^2\)) helps significantly in easing the solving process. With practice, you'll get better at spotting how to manipulate the bases in exponential equations.

Base Conversion

Base conversion, especially in logarithms, involves rewriting numbers so that their bases match or are convenient to work with. This allows you to solve for exponents more easily. For example, converting large numbers into powers of smaller numbers, like rewriting 100 as \(10^2\), can often transform a problem into a simpler version of itself. This is exactly what we did when solving the exercise at hand. When faced with a logarithmic or exponential problem:

• Look for opportunities to break down any base into smaller components.
• Check if a number can be expressed as a power of another number you already have in the equation.
• Use these conversions to simplify the equation and make solving it more manageable.

Through practice, identifying when and how to convert bases will become second nature. This trick is particularly useful in ensuring you can equate and solve exponents correctly.

Exponents

Exponents are a fundamental part of mathematics, allowing us to express large numbers compactly and explore the patterns and rules of multiplication. In the realm of logarithms, understanding exponents is key because a logarithm essentially asks "what exponent do we need?". Key points to remember about exponents include:

• \(a^m \times a^n = a^{m+n}\). When you multiply like bases, you add the exponents.
• \((a^m)^n = a^{mn}\). When you raise a power to another power, you multiply the exponents.
• \(a^{-n} = \frac{1}{a^n}\). A negative exponent represents a reciprocal.

Understanding these properties helps tremendously when solving equations where the variable is an exponent, including converting log equations into exponential ones as we did in the provided exercise.