1. 1/3log base 8 of (x+1)=2log base 8 of 3-(2/3)log base 8 of (x+1)
2. 2^x+8 times 2^=x all over 2 = 3
3. if log base a of 3= x and log base a of 2 = y, find each of thefollowing in terms of x and y
log base a (18a^3)
thanks!!
1.
omitting all the base 8 stuff, we have
1/3 log(x+1) = 2log3 - 2/3 log(x+1)
log(x+1) = 2log3
log(x+1) = log9
x+1=9
x=8
2.
not sure what the extra = means.
(2^x + 8*2^x)/2 = 3
9*2^x/2 = 3
9*2^x = 6
2^x = 3/2
x = ln(3/2)/ln2
May be a typo here; it's an odd answer
3.
omitting all the base a stuff,
log(18x^3) = log18 + loga^3
= log9+log2+3loga
= 2log3 + log2 + 3loga
= 2x+y+3
oops it was 2^-x
im confused why would you omit the bases?
just for readability. Got tired of typing it in. And, until you need to resolve the base, it doesn't really matter. In #1, the equations work regardless of base, since we are working with logs on both sides of the equation.
In #3, the base didn't matter until we could use it to say that log_a(a) = 1
For #2, I suspected it might have been 2^-x, and it helps a lot!
(2^x + 8*2^-x)/2 = 3
2^x + 8/2^x = 6
if you let u = 2^x, then you have
u+8/u = 6
u^2 + 8 = 6u
u^2 - 6u + 8 = 0
(u-4)(u-2) = 0
u=4 or u=2
2^x=4 or 2^x=2
x=2 or x=1
1. To solve the equation 1/3log base 8 of (x+1) = 2log base 8 of 3 - (2/3)log base 8 of (x+1), we can simplify and solve step by step:
First, we can simplify the equation:
1/3log base 8 of (x+1) = 2log base 8 of 3 - (2/3)log base 8 of (x+1)
Multiply both sides by 3 to eliminate the fraction:
log base 8 of (x+1) = 6log base 8 of 3 - 2log base 8 of (x+1)
Now, we can combine similar terms:
log base 8 of (x+1) + 2log base 8 of (x+1) = 6log base 8 of 3
Simplify:
3log base 8 of (x+1) = 6log base 8 of 3
Divide both sides by 3:
log base 8 of (x+1) = 2log base 8 of 3
Now, we can rewrite the equation in exponential form:
8^(log base 8 of (x+1)) = 8^(2log base 8 of 3)
(x+1) = (8^2) * 3
(x+1) = 64 * 3
(x+1) = 192
x = 192 - 1
x = 191
Therefore, the solution to the equation is x = 191.
2. To solve the equation 2^x+8 times 2^=x all over 2 = 3, we can simplify and solve step by step:
First, we need to clarify the equation. It seems that there is a missing exponent after the second 2^. Let's assume it is 8.
The equation becomes 2^(x+8) * 2^8 / 2 = 3.
Next, let's simplify using the properties of exponents:
2^(x+8+8-1) = 3.
Now, we can rewrite the equation:
2^(x+15) = 3.
To solve for x, we can take the logarithm of both sides of the equation. Let's use logarithm base 2 since the base of the original equation is 2:
log base 2 (2^(x+15)) = log base 2 (3).
By applying the power rule of logarithms, we get:
(x+15) = log base 2 (3).
Finally, we solve for x by subtracting 15 from both sides of the equation:
x = log base 2 (3) - 15.
Therefore, the solution to the equation is x = log base 2 (3) - 15.
3. To find log base a (18a^3) in terms of x and y, we can use the properties of logarithms:
First, recall that log base a of (mn) = log base a of m + log base a of n.
We can rewrite 18a^3 as (2 * 9 * a * a^2) and use the above property of logarithms:
log base a (18a^3) = log base a (2 * 9 * a * a^2).
Next, we can break down each term under the logarithm:
log base a (2) + log base a (9) + log base a (a) + log base a (a^2).
Now, we can simplify further:
log base a (2) + log base a (9) + log base a (a) + 2log base a (a).
By the given information, we know that log base a of 3 = x and log base a of 2 = y. We can substitute these values into the equation:
x + 2y + 1 + 2(1).
Simplifying further, we get:
x + 2y + 1 + 2.
Therefore, log base a (18a^3) in terms of x and y is x + 2y + 3.