Express as a single logarithm.
1/2log(subscript)3 144-log(subscript)3 4+2log(subscript)3 3
Please Help!I just can't seem to get the correct answer when I do it.
log3 (144)^(1/2) = log3 (12)
2 log3(3) = log3(9)
so we have
log3 [ 12*9/4 ]
log3 [ 3^3 ]
3 log3{3)
3
Thank You, it really helped me.
To express the expression as a single logarithm, you can use the logarithmic properties. Let's break down the given expression step by step:
1/2log(subscript)3 144 - log(subscript)3 4 + 2log(subscript)3 3
First, let's simplify each term individually:
1/2log(subscript)3 144 can be rewritten using the power rule for logarithms:
log(subscript)3 (144)^(1/2)
Next, log(subscript)3 4 remains the same.
For 2log(subscript)3 3, we can use the power rule for logarithms to rewrite it as:
log(subscript)3 (3^2)
Now, we can rewrite the expression as:
log(subscript)3 (144)^(1/2) - log(subscript)3 4 + log(subscript)3 (3^2)
Using the logarithmic property of subtraction, we can combine the second and third terms into a single term:
log(subscript)3 [(144)^(1/2) / 4 * 3^2]
Now let's simplify further:
log(subscript)3 [(12)^(1/2) / 4 * 9]
Using the logarithmic property of division, we can rearrange the terms:
log(subscript)3 [(12)^(1/2)] - log(subscript)3 [4 * 9]
Taking the square root of 12:
log(subscript)3 [√12] - log(subscript)3 [4 * 9]
Simplifying further:
log(subscript)3 [√12] - log(subscript)3 [36]
Using the logarithmic property of subtraction again:
log(subscript)3 [√12 / 36]
However, the expression can still be simplified further. We can rewrite 36 as 6^2:
log(subscript)3 [√12 / 6^2]
Using the logarithmic property of division again:
log(subscript)3 [√12 / 6^2] = log(subscript)3 [√12] - log(subscript)3 [6^2]
Finally, we have expressed the given expression as a single logarithm:
log(subscript)3 [√12] - log(subscript)3 [6^2]