Question

Если log(3)4=a и log(5)4=b то найдите log(9)100

Answer 1

$log_34=a\; \; ,\; \; log_54=b\\\\\\log_34=log_32^2=2log_32=a\; \; \to \; \; log_32=\frac{a}{2}\; ,\; log_23=\frac{1}{log_32}=\frac{2}{a}\\\\log_54=log_52^2=2log_52=b\; \; \to \; \; log_52=\frac{b}{2}\; ,\; \; log_25=\frac{1}{log_52}=\frac{2}{b}\\\\\\log_9100=log_{3^2}(5\cdot 2)^2=\frac{1}{2}\cdot 2\cdot log_3(5\cdot 2)=log_35+log_32=\frac{log_25}{log_23}+\frac{a}{2}=\\\\=\frac{2/b}{2/a}+\frac{a}{2}=\frac{a}{b}+\frac{a}{2}=\frac{a(2+b)}{2b}$

Answer 2

To find the value of log(9)100, we can use logarithm properties to rewrite it in terms of a and b.

Step 1: Express 100 as a power of 9 and 4: 100 = 10^2 = (9*4)^2 = 9^2 * 4^2

Step 2: Use the property of logarithms: log_b(xy) = log_b(x) + log_b(y)

log(9)100 = log(9)(9^2 * 4^2) = log(9)(9^2) + log(9)(4^2)

Step 3: Use the property of logarithms: log_b(b^a) = a

log(9)100 = 2 + log(9)(4^2)

Step 4: Use the given information log(3)4=a and log(5)4=b:

Since 4 = 2^2, we can express log(9)(4^2) as log(9)(2^(2*2)).

Now, use the property of logarithms: log_b(x^a) = a * log_b(x)

log(9)(4^2) = 2 * log(9)2

Step 5: Substituting the given information:

log(9)100 = 2 + 2 * log(9)2

Now, we need to express log(9)2 in terms of a and b.

Step 6: Use the change of base formula for logarithms:

log(9)2 = log(2)2 / log(2)9

We know that log(3)4 = a and log(5)4 = b.

Step 7: Express log(2)9 in terms of a and b:

log(2)9 = log(2)(3^2) = 2 * log(2)3

Step 8: Express log(2)3 in terms of a:

log(2)3 = log(2)(2^a) = a * log(2)2

Step 9: Substitute the value of log(2)3 into log(2)9:

log(2)9 = 2 * log(2)3 = 2 * (a * log(2)2) = 2a * log(2)2

Step 10: Substitute log(2)9 back into the expression for log(9)2:

log(9)2 = log(2)2 / log(2)9 = log(2)2 / (2a * log(2)2) = 1 / (2a)

Step 11: Substitute log(9)2 back into the expression for log(9)100:

log(9)100 = 2 + 2 * log(9)2 = 2 + 2 * (1 / (2a)) = 2 + 1 / a

So, the final answer is log(9)100 = 2 + 1 / a.