I don't understand how to do these w/o calc.
I tried to write it in a way that will make someone understand how to read it. Hope I typed it clear enough.Thanks so much for the help anyone!
How to find the exact value of logarithm:
10. log5^100 -log5^4
11. log4^8 +log4^8
12. log squareroot 100
13. log4^40 -log4^10
19. log 3^20.25 +log3^4
22. 7 log2 ^16 +log2 (1/16)
27. log 2 +log 5-log 10
31. 2 ln e^6 -ln e^5
36. log 100 + log 10,000
log5 (25)
5^(log5(25) = 25
but
5^2 = 25
so
log5(25) = 2
you must memorize the 3 main rules of logs
1. log (AB) = log A + log B
2. log (A/B) = logA - logB
3 log (A^n) = nlogA
for any base of the log as long as the base is the same throughout the equation
e.g.
#22
7log 2^16 + log (2^(1/16)) , I assumed a typo here
= 7(16)log 2 + (1/16)log 2
= (1793/16) log 2
#11
log 4^8 + log4^8
= log (4^8 * 4^8)
= log 4^16
or
= 16 log4
To find the exact value of logarithms without using a calculator, we can rely on certain properties of logarithms.
1. log(a) - log(b) = log(a/b): This property states that the difference between two logarithms with the same base is equal to the logarithm of their quotient.
2. log(a) + log(b) = log(a * b): This property states that the sum of two logarithms with the same base is equal to the logarithm of their product.
3. log(x^n) = n * log(x): This property states that the logarithm of a number raised to a power is equal to the product of that power and the logarithm of the number.
4. log(x) = a is equivalent to x = base^a: This property states that a logarithmic equation is equivalent to an exponential equation where the base is raised to the power of the logarithm.
Using these properties, let's evaluate each of the given logarithmic expressions:
10. log5^100 - log5^4 = log(5^100/5^4) = log(5^(100-4)) = log(5^96)
11. log4^8 + log4^8 = log(4^8 * 4^8) = log(4^(8+8)) = log(4^16)
12. log(sqrt(100)) = log(10)
13. log4^40 - log4^10 = log(4^40/4^10) = log(4^(40-10)) = log(4^30)
19. log(3^20.25) + log(3^4) = log(3^(20.25+4)) = log(3^24.25)
22. 7log2^16 + log2(1/16) = 7log(2^16) + log(2^(1/16))
27. log(2) + log(5) - log(10) = log(2*5/10) = log(1) = 0
31. 2ln(e^6) - ln(e^5) = ln(e^(2*6)/e^5) = ln(e^12/e^5) = ln(e^7)
36. log(100) + log(10000) = log(100 * 10000) = log(1000000)
Please note that some of these expressions might simplify further, depending on specific values and numerical computations.