QuestionCari nilai $h$ bagi $5 \log _{3}\left(\frac{h}{3}\right)+\log _{3} \sqrt{h}=6$ .

Studdy Solution

STEP 1

Assumptions1. We are given the equation $5 \log{3}\left(\frac{h}{3}\right)+\log{3} \sqrt{h}=6$ . . We need to find the value of $h$ that satisfies this equation.
3. We assume that $h >0$ because the argument of a logarithm and the radicand of a square root must be positive.

STEP 2

First, let's simplify the equation by using the properties of logarithms. We can start by applying the power rule of logarithms to the first term on the left-hand side. The power rule states that $a \log_b c = \log_b (c^a)$ .
$5 \log{}\left(\frac{h}{}\right) = \log{}\left(\left(\frac{h}{}\right)^5\right)$

STEP 3

Now, apply the power rule to the second term on the left-hand side. The power rule states that $\log_b \sqrt{c} = \frac{1}{2} \log_b c$ .
$\log{3} \sqrt{h} = \frac{1}{2} \log{3} h$

STEP 4

Substitute the simplified terms back into the original equation.
\log{3}\left(\left(\frac{h}{3}\right)^\right) + \frac{1}{2} \log{3} h =6

STEP 5

We can further simplify the equation by combining the logarithms on the left-hand side into a single logarithm. The product rule of logarithms states that $\log_b a + \log_b c = \log_b (a \cdot c)$ .
$\log{3}\left(\left(\frac{h}{3}\right)^5 \cdot h^{1/2}\right) =$

STEP 6

Next, we can eliminate the logarithm by raising both sides of the equation to the base of the logarithm, which is3. This gives us $\left(\frac{h}{3}\right)^5 \cdot h^{1/2} =3^6$

STEP 7

implify the right-hand side of the equation.
$\left(\frac{h}{3}\right)^5 \cdot h^{1/2} =729$

STEP 8

Now, rewrite the left-hand side of the equation to make it easier to solve for $h$ .
$\frac{h^5}{3^5} \cdot h^{1/2} =729$

STEP 9

Multiply both sides of the equation by $3^5$ to get rid of the denominator on the left-hand side.
$h^5 \cdot h^{/2} =729 \cdot3^5$

STEP 10

implify the right-hand side of the equation.
$h^5 \cdot h^{/2} =59049$

STEP 11

Combine the powers of $h$ on the left-hand side of the equation.
$h^{5 +/} =59049$

STEP 12

implify the exponent on the left-hand side of the equation.
$h^{11/2} =59049$

STEP 13

To solve for $h$ , raise both sides of the equation to the power of $2/11$ .
$h = (59049)^{2/11}$

STEP 14

Calculate the value of $h$ .
$h =9$ The value of $h$ that satisfies the given equation is9.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms