Math  /  Algebra

QuestionSolve the following exponential equation. Express your answer as both an exact expression and a decimal approximation rounded to two decimal place 8x+9=1088^{x+9}=108

Studdy Solution

STEP 1

What is this asking? We need to find the value of xx that makes 88 raised to the power of x+9x + 9 equal to 108108.
Then, we need to give the answer in two ways: exact and rounded to two decimal places. Watch out! Remember, when dealing with exponential equations, it's easy to get tripped up with the order of operations.
Don't add before you exponentiate!

STEP 2

1. Isolate the exponential term
2. Apply the logarithm
3. Solve for xx
4. Approximate the solution

STEP 3

We already have the exponential term, 8x+98^{x+9}, isolated on one side of the equation: 8x+9=1088^{x+9} = 108.
So, we're good to go!
On to the next step!

STEP 4

To get the variable xx out of the exponent, we can **apply the logarithm** to both sides of the equation.
Let's use the **natural logarithm (ln)**, because it's super handy!
Remember, applying the same operation to both sides of an equation keeps it balanced. ln(8x+9)=ln(108) \ln(8^{x+9}) = \ln(108)

STEP 5

Now, we can use the **power rule of logarithms**.
This rule says ln(ab)=bln(a)\ln(a^b) = b \cdot \ln(a).
This lets us bring the exponent down in front: (x+9)ln(8)=ln(108) (x+9) \cdot \ln(8) = \ln(108)

STEP 6

To **isolate** the term with xx, we'll **divide both sides** by ln(8)\ln(8): (x+9)ln(8)ln(8)=ln(108)ln(8) \frac{(x+9) \cdot \ln(8)}{\ln(8)} = \frac{\ln(108)}{\ln(8)} x+9=ln(108)ln(8) x + 9 = \frac{\ln(108)}{\ln(8)}

STEP 7

Almost there!
We just need to **subtract** 99 from both sides to get xx by itself: x+99=ln(108)ln(8)9 x + 9 - 9 = \frac{\ln(108)}{\ln(8)} - 9 x=ln(108)ln(8)9 x = \frac{\ln(108)}{\ln(8)} - 9 This is our **exact answer**!

STEP 8

Now, let's use a calculator to find the **decimal approximation**. x4.68212.07949 x \approx \frac{4.6821}{2.0794} - 9 x2.25179 x \approx 2.2517 - 9 x6.7483 x \approx -6.7483

STEP 9

Rounding to two decimal places gives us: x6.75 x \approx -6.75 This is our **approximate answer**!

STEP 10

Exact solution: x=ln(108)ln(8)9x = \frac{\ln(108)}{\ln(8)} - 9 Approximate solution: x6.75x \approx -6.75

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