QuestionSolve for $x$ in the following exponential equation by matching the bases. $\left(\frac{1}{3}\right)^{-3 x}-81^{(x-1)}=0$

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes this equation true! Watch out! Don't forget those exponent rules!
Also, remember that fractions can be tricky!

STEP 2

1. Rewrite the Equation
2. Match the Bases
3. Combine and Solve

STEP 3

First, let's isolate one of the terms with $x$ by adding $81^{(x-1)}$ to both sides of the equation.
This gives us: $\left(\frac{1}{3}\right)^{-3x} = 81^{(x-1)}$ Why did we do this?
It's like setting the stage before the main act!
Now we can work on making the bases the same.

STEP 4

Remember that $\frac{1}{3}$ is the same as $3^{-1}$ .
So, we can rewrite the left side of the equation as: $(3^{-1})^{-3x} = 81^{(x-1)}$ This is a super helpful trick because we want to get the same base on both sides of the equation!

STEP 5

Now, we can use the power of a power rule.
Remember, when you have an exponent raised to another exponent, you multiply them!
So, $(-1) \cdot (-3x)$ becomes $3x$ .
This gives us: $3^{3x} = 81^{(x-1)}$ We're getting closer to matching those bases!

STEP 6

We know that $81$ is $3$ multiplied by itself four times, or $3^4$ .
Let's substitute that into our equation: $3^{3x} = (3^4)^{(x-1)}$ Look at that!
Both sides have a base of 3!

STEP 7

Let's use that power of a power rule again on the right side.
We multiply $4$ by $(x-1)$ to get $4x - 4$ .
So, our equation becomes: $3^{3x} = 3^{4x-4}$ Matching bases makes solving for $x$ so much easier!

STEP 8

Since the bases are the same, the exponents must also be equal.
This gives us a nice, simple equation: $3x = 4x - 4$ We're in the home stretch now!

STEP 9

Let's subtract $4x$ from both sides: $3x - 4x = 4x - 4 - 4x$ $-x = -4$ Now, we can multiply both sides by $-1$ to get $x$ by itself: $(-1) \cdot (-x) = (-1) \cdot (-4)$ $x = 4$ We did it!
We found the value of $x$ !

STEP 10

$x = 4$

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