QuestionSolve for $x$ , rounding to the nearest hundredth. $30 \cdot 3^{x}=180$

Studdy Solution

STEP 1

What is this asking? We're trying to find the power of 3 that, when multiplied by 30, gives us 180! Watch out! Don't forget the order of operations – we need to deal with that 30 multiplying the exponential term before we can work with the exponent itself.

STEP 2

1. Isolate the Exponential Term
2. Apply the Logarithm
3. Calculate and Round

STEP 3

We want to get that $3^x$ by itself first.
Since it's being multiplied by 30, we'll divide both sides of the equation by 30! $\frac{30 \cdot 3^x}{30} = \frac{180}{30}$

STEP 4

This simplifies to: $3^x = 6$ Now we're getting somewhere!
We've isolated the exponential term, which is the key to unlocking this problem.

STEP 5

To get that $x$ down from the exponent, we're going to use a logarithm.
Let's use the base-10 logarithm (log) because it's easily available on most calculators.
Taking the log of both sides gives us: $\log(3^x) = \log(6)$

STEP 6

Now, we can use the super handy power rule of logarithms, which says $\log(a^b) = b \cdot \log(a)$ .
This lets us bring that $x$ down in front: $x \cdot \log(3) = \log(6)$ Almost there!

STEP 7

To isolate $x$ , we just need to divide both sides by $\log(3)$ : $x = \frac{\log(6)}{\log(3)}$

STEP 8

Now, grab your calculator and crunch those numbers: $x \approx \frac{0.77815}{0.47712}$ $x \approx 1.6309$

STEP 9

Finally, we need to round our answer to the nearest hundredth, which gives us: $x \approx 1.63$ Boom! We did it!

STEP 10

$x \approx 1.63$

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