Problem 35 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation for \(x.\) $$\log _{8} x=\log _{8} 10^{-1}$$

Step-by-Step Solution

Answer

The solution to the given logarithmic equation is \(x = 0.1\).

Step 1: Understanding the Logarithmic Equation

Take a look at the given equation: \(\log _{8} x=\log _{8} 10^{-1}\). This is a logarithmic equation, and the bases on both sides are the same (which is 8). Therefore, we can directly equate x with the argument of the logarithm on the right side.

Step 2: Applying Logarithmic Property

The property of logarithms states that if \(\log_b a = \log_b c\), then \(a = c\). Therefore, we can equate \(x\) with \(10^{-1}\). Thus, \(x = 10^{-1}\).

Step 3: Simplifying the Result

Simplify \(10^{-1}\) to get the solution in decimal form. \(10^{-1}\) is exactly the same as 0.1. Thus, \(x = 0.1\).

Key Concepts

Logarithm Properties

Logarithms are like detectives of the math world; they help us figure out the mystery number called the exponent. One key property that they use is that if two logs with the same base are equal, then their arguments must be equal too. This is really handy when solving equations because it means we can drop the log part and just set the insides—the arguments—equal to each other.

For example, in the exercise, we have \[\log_{8} x = \log_{8} 10^{-1}\]. Because the logs on both sides have the same base of 8, we can just say \[x = 10^{-1}\], which is way simpler to solve. This property makes logarithms a valuable tool in our mathematical toolkit and shows why log properties are so essential to grasp for solving equations.

Equations with Same Base

When we're working with equations that have the same base, like in our exercise where both sides have a log base 8, it's like finding two puzzle pieces that fit perfectly together. This similarity lets us make a shortcut and skip some steps. Normally, solving logarithmic equations involves changing to exponential form, but with a matching base, we don't need to do that.

Once we realize they have the same base, all we have to do is compare what's inside the logs, which in math we call the 'arguments'. So, for the given equation \(\log _{8} x = \log _{8} 10^{-1}\), the base 8 just falls away like magic and we're left with \(x = 10^{-1}\). It's like getting the answer to a riddle without having to solve the whole puzzle!

Simplifying Exponents

Playing with exponents is a bit like baking; you have to know the right amounts to use. Simplifying exponents means writing them in the simplest form possible. Think of it this way: exponents are telling you how many times to multiply a number by itself. A negative exponent, like \(10^{-1}\), is the recipe for making a fraction. It tells you that instead of multiplying, you divide 1 by the number raised to the positive exponent.

So, when you see \(10^{-1}\), you can almost hear it saying, 'Easy peasy! Just flip me over and you'll find that I'm really 1 divided by 10, which is 0.1'. By simplifying exponents, everything becomes easier to digest, and our final solution for \(x\) is just a simple, sweet number: 0.1.