Find the exact value without a calc
1. log base 3 of 8 times log base 8 of 9. I started by changing both bases to 10 but don't know what to do from there.
2. e ^ log base e^2^9. I hope that isnt confusing. My teacher said we are suppose to get the bases all e, but how because there is only 1 log.
Suppose that ln 2= a and ln 3= b. Use the properties of logs to write each in terms of a and b
1. ln .5
thanks
recall that
log_a(n) = log_b(n)/log_a(b)
log_a(n) = 1/log_n(a)
1.
log_3(8) * log_8(9)
= 1/log_8(3) * log_8(9)
= log_3(9)
= 2
2.
recall that
e^ln(n) = n, so
e^ln(2^9) = 2^9 = 512
3.
.5 = 1/2, so
ln .5 = ln(2^-1) = -ln(2) = -a
To find the exact value of a logarithmic expression without using a calculator, you can use the properties of logarithms and simplify the expression step by step.
1. log base 3 of 8 times log base 8 of 9:
Let's break it down step by step:
Using the change of base formula, you can rewrite the logarithms in terms of the common logarithm (log base 10).
log base 3 of 8 can be written as log(8)/log(3)
log base 8 of 9 can be written as log(9)/log(8)
Now, simplify the expression:
log(8)/log(3) * log(9)/log(8)
The log(8) in the numerator and denominator will cancel out:
log(9)/log(3)
Finally, evaluate the expression:
log(9)/log(3) ≈ 2.0801
2. e ^ log base e^2^9:
For this expression, you can use the property of logarithms:
e ^ log base a of b = b
In this case, a = e and b = e^2^9.
So, e ^ log base e^2^9 = e^2^9
Now, evaluate the expression:
e^2^9 ≈ 9.99982895704e+78 (approximately)
For the second part of your question:
To write ln 0.5 in terms of a and b, you can use the property of logarithms:
ln (ab) = ln a + ln b
ln (0.5) = ln (2^-1)
Using the property of logarithm for exponents:
ln (2^-1) = -ln 2
Since ln 2 = a:
-ln 2 = -a
Therefore, ln 0.5 = -a.