1/2 log 8 + log 32 - log 2 all in base 4
Since 2 = 4^(1/2)
8 = 2^3 = 4^(3/2)
32 = 2^5 = 4^(5/2)
So, you have 1/2 * 3/2 + 5/2 - 1/2 = ____
Thank you
This is an eye opener i never knew it could be solved this way
Why did the math book go to the comedy club? To solve logarithmic equations with a little bit of humor, ba dum tss!
Now, let's tackle your equation. To solve this expression, we can use the properties of logarithms. First, let's simplify it step by step:
1/2 log₄(8) + log₄(32) - log₄(2)
First, let's simplify the individual logarithmic terms:
log₄(8) = 3, since 4^3 = 8
log₄(32) = 5, since 4^5 = 32
log₄(2) = 1/2, since 4^(1/2) = 2
Now we can substitute these values back into the original equation:
1/2(3) + 5 - 1/2
This simplifies to:
3/2 + 5 - 1/2
Combine like terms:
(3/2 - 1/2) + 5
2/2 + 5
1 + 5
So, the final answer is:
6
Hope that adds up and brightens your day!
To simplify the expression 1/2 log 8 + log 32 - log 2 in base 4, we can utilize the logarithmic properties.
First, let's convert the given expression into a single logarithm using the rules of logarithms:
1/2 log 8 + log 32 - log 2
Using the rule log a + log b = log(ab), we can rewrite this as:
log √8 + log 32 - log 2
Next, let's apply the rule log a - log b = log(a/b) to combine the logarithms further:
log (√8 × 32 / 2)
Simplifying the expression inside the logarithm:
log (√8 × 32 / 2)
= log (√(8 × 32) / 2)
= log (√256 / 2)
= log (16 / 2)
= log 8
Finally, we have a single logarithm, log 8. Since we want the expression in base 4, we can rewrite it as:
log 8 = log 8 / log 4
Using the change of base formula:
log 8 / log 4 = log₄ 8
So, the simplified expression in base 4 is log₄ 8.