If log3 𝑎 + 2 = 3log3 𝑏 , express 𝑎 in terms of 𝑏.

Question

If log3 𝑎 + 2 = 3log3 𝑏 , express 𝑎 in terms of 𝑏.

A. 𝑎 = 𝑏3
9
B.𝑎 = 𝑏3 − 9
C.𝑎 = 9𝑏3
D. 𝑎 = 𝑏3 − 3

Answer 1

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.

Answer 2

I apologize for the confusion. You are correct. I will clarify that the answer is 𝑎 = 𝑏3/9, not 𝑎 = 𝑏/3. Thank you for bringing it to my attention.

Answer 3

To express 𝑎 in terms of 𝑏, we need to solve the equation log3 𝑎 + 2 = 3log3 𝑏.

Step 1: Move the terms involving 𝑎 to one side of the equation and the terms involving 𝑏 to the other side of the equation.

Subtract 3log3 𝑏 from both sides:
log3 𝑎 + 2 - 3log3 𝑏 = 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 𝑥 + log 𝑦 = log(𝑥𝑦).

log3 𝑎 - 3log3 𝑏 + 2 = 0

Step 3: Apply the logarithmic property to combine the logarithms.

Using log 𝑥 - log 𝑦 = log(𝑥/𝑦), we can write the equation as:

log3 (𝑎/𝑏^3) + 2 = 0

Step 4: Rewrite the equation in exponential form.

Since log3 (𝑎/𝑏^3) + 2 = 0, we can rewrite it as:

3^0 = 𝑎/𝑏^3

Simplifying 3^0 as 1, we get:

1 = 𝑎/𝑏^3

Step 5: Multiply both sides of the equation by 𝑏^3.

𝑏^3 = 𝑎/𝑏^3 * 𝑏^3

The 𝑏^3 on the right side cancel out, leaving us with:

𝑏^3 = 𝑎

Therefore, the expression for 𝑎 in terms of 𝑏 is 𝑎 = 𝑏^3.

Hence, the correct option is A. 𝑎 = 𝑏^3.

Answer 4

To express 𝑎 in terms of 𝑏, we need to isolate 𝑎 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: logₐ(𝑏) + logₐ(𝑐) = logₐ(𝑏𝑐).
Applying this property, we can rewrite the equation as:
log₃(𝑎) + log₃(2) = log₃(𝑏³).

Step 2: Combine the terms on the left side of the equation.
Using another property of logarithms, we know that logₐ(𝑏) + logₐ(𝑐) = logₐ(𝑏𝑐), we can combine the terms on the left side:
log₃(2𝑎) = log₃(𝑏³).

Step 3: Remove the logarithm from the equation.
To remove the logarithm, we can apply the property: logₐ(𝑏) = 𝑐 is equivalent to 𝑎 = 𝑏ᶜ.
Using this property, we can rewrite the equation as:
2𝑎 = 𝑏³.

Step 4: Isolate 𝑎.
To isolate 𝑎, we need to divide both sides of the equation by 2:
𝑎 = 𝑏³/2.

So, the expression for 𝑎 in terms of 𝑏 is 𝑎 = 𝑏³/2.

None of the answer options provided match the expression we obtained.

Answer 5

Using the laws of logarithms, we can rewrite the equation as:

log3 𝑎 + log3 3^2 = log3 𝑏^3

Combining the logs on the left:

log3 (𝑎⋅3^2) = log3 𝑏^3

Setting the arguments equal:

𝑎⋅3^2 = 𝑏^3

Simplifying:

𝑎 = 𝑏^3/9

Therefore, the answer is A. 𝑎 = 𝑏3/9 or 𝑎 = 𝑏3.