Solve each logarithmic equation. logx+log(x-48)=2

Name: Problem 59, Chapter 4: College Algebra
Uploaded: 2025-01-22T17:49:36-08:00
Channel: Claire Rochford
Description: Solve each logarithmic equation. logx+log(x-48)=2

Key Concepts

Solving Quadratic Equations

After transforming the logarithmic equation into an algebraic one, the result can sometimes be a quadratic equation. Being proficient in techniques for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula, is essential for finding the correct solutions.

Conversion Between Logarithmic and Exponential Forms

This concept involves rewriting logarithmic equations in their equivalent exponential form based on the definition of a logarithm (if log_b(y) = c, then y = b^c). Converting the equation allows one to use algebraic techniques to solve for the unknown variable.

Properties of Logarithms

This concept involves understanding the rules that govern logarithmic operations, such as the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the factors. These properties allow for the simplification of expressions involving the addition or subtraction of logarithms, making it easier to isolate variables within logarithmic equations.

Domain Restrictions

In logarithmic functions, the argument of the logarithm must be positive. This key concept is crucial because any potential solution that results in a non-positive argument is not valid. Ensuring that the domain restrictions are satisfied is an essential step when solving logarithmic equations.

Transcript

00:01 Okay, we are told to solve this logarithmic equation.

00:05 The first thing i look for when i'm solving a logarithmic or exponential equation is if there is a log on one side or both sides.

00:14 And in this case, there's only logs on this left -hand side, which means that the only way we can get rid of the log is by undoing it with an inverse.

00:22 We can't undo a log with an inverse, which would be an exponential, unless we just have one log.

00:29 And right now we have two different logs.

00:31 So we're going to want to condense these guys.

00:34 That word condense hopefully should bring up some memories of log rules.

00:40 We see addition between these two logs.

00:43 So just as a reminder, the log rule that we do when we have logs being added together is we can put the two insides into one log where they are multiplying together.

00:55 So that's the first thing we're going to do here is we're going to use this log property.

00:59 So we're going to have one log with our first inside times our second inside.

01:08 And i'm going to go ahead and simplify this.

01:10 I'm going to distribute that just to make it easier on myself.

01:13 It looks a little bit weird right now.

01:14 So if i distribute that x, i get x squared minus 48x.

01:21 So my next step is to get rid of the log entirely so that our x is not trapped inside.

01:28 Right now we can't do too much because our x is stuck inside of a log.

01:32 We need to get rid of it.

01:35 And the way we get rid of it is with an exponential of a matching base.

01:40 It's kind of like undoing multiplication with division and undoing squares with square roots and so forth.

01:47 The way we get rid of a log is with an exponential.

01:50 We just have to make sure that it's the same base.

01:53 Right now we don't see any base.

01:55 Typically, you find them down by the bottom of your g.

01:58 If we see the word log with no base, we can assume it's a 10.

02:02 It's kind of our invisible base.

02:05 So what we're going to do is we're going to write that base really big on both sides of our equal sign.

02:11 And then we're going to copy down what we originally had on both sides up tiny in the corner.

02:24 And the reason we do this is because exponential base 10 and log base 10 cancel each other out.

02:31 And that's going to leave us with just the inside of that exponential.

02:38 So x squared minus 48x...

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