If log3 ๐ + 2 = 3log3 ๐ , express ๐ in terms of ๐.
A. ๐ = ๐3
9
B.๐ = ๐3 โ 9
C.๐ = 9๐3
D. ๐ = ๐3 โ 3
you should have stopped at a = b^3/9
That is NOT the same as b/3 !! Maybe you should not write "b3" if you mean b^3.
I apologize for the confusion. You are correct. I will clarify that the answer is ๐ = ๐3/9, not ๐ = ๐/3. Thank you for bringing it to my attention.
To express ๐ in terms of ๐, we need to solve the equation log3 ๐ + 2 = 3log3 ๐.
Step 1: Move the terms involving ๐ to one side of the equation and the terms involving ๐ to the other side of the equation.
Subtract 3log3 ๐ from both sides:
log3 ๐ + 2 - 3log3 ๐ = 0
Step 2: Combine the terms on the left side of the equation.
We can combine the two logarithms using the logarithmic property that states log ๐ฅ + log ๐ฆ = log(๐ฅ๐ฆ).
log3 ๐ - 3log3 ๐ + 2 = 0
Step 3: Apply the logarithmic property to combine the logarithms.
Using log ๐ฅ - log ๐ฆ = log(๐ฅ/๐ฆ), we can write the equation as:
log3 (๐/๐^3) + 2 = 0
Step 4: Rewrite the equation in exponential form.
Since log3 (๐/๐^3) + 2 = 0, we can rewrite it as:
3^0 = ๐/๐^3
Simplifying 3^0 as 1, we get:
1 = ๐/๐^3
Step 5: Multiply both sides of the equation by ๐^3.
๐^3 = ๐/๐^3 * ๐^3
The ๐^3 on the right side cancel out, leaving us with:
๐^3 = ๐
Therefore, the expression for ๐ in terms of ๐ is ๐ = ๐^3.
Hence, the correct option is A. ๐ = ๐^3.
To express ๐ in terms of ๐, we need to isolate ๐ on one side of the equation. Let's break down the steps to solve the equation:
Step 1: Use properties of logarithms to simplify the equation.
Since we have the sum in the left part, we can use the property of logarithms that states: logโ(๐) + logโ(๐) = logโ(๐๐).
Applying this property, we can rewrite the equation as:
logโ(๐) + logโ(2) = logโ(๐ยณ).
Step 2: Combine the terms on the left side of the equation.
Using another property of logarithms, we know that logโ(๐) + logโ(๐) = logโ(๐๐), we can combine the terms on the left side:
logโ(2๐) = logโ(๐ยณ).
Step 3: Remove the logarithm from the equation.
To remove the logarithm, we can apply the property: logโ(๐) = ๐ is equivalent to ๐ = ๐แถ.
Using this property, we can rewrite the equation as:
2๐ = ๐ยณ.
Step 4: Isolate ๐.
To isolate ๐, we need to divide both sides of the equation by 2:
๐ = ๐ยณ/2.
So, the expression for ๐ in terms of ๐ is ๐ = ๐ยณ/2.
None of the answer options provided match the expression we obtained.
Using the laws of logarithms, we can rewrite the equation as:
log3 ๐ + log3 3^2 = log3 ๐^3
Combining the logs on the left:
log3 (๐โ
3^2) = log3 ๐^3
Setting the arguments equal:
๐โ
3^2 = ๐^3
Simplifying:
๐ = ๐^3/9
Therefore, the answer is A. ๐ = ๐3/9 or ๐ = ๐3.