a) b) c) d) e) f) g) h).... Question from @chinczikan

a) ... = log3 (log10^3) = log3 (3 * log10) = log 3 (3 * 1) = log3 3 = 1

b) ... = 10 * 10^(log (3/2)) = 10 * 3/2 = 15

c) .... = log1/2 (log3 3^2) = log1/2 (2 * log3 3) = log1/2 (2 * 1) = log1/2 2 = log1/2 (1/2)^(-1) = (-1) * log1/2 1/2 = (-1) * 1 = - 1

d) ... = 10^(log10^2 + log8) = 10^(log100 + log8) 10^log(100 * 8) = 10^log800 = 800

log5 6^2 * log5 8^2 2* log5 6 * 2 * log5 8 4

e) .... = -----------------------=-------------------------- = -----

log5 6 * log5 8^3 log5 6 * 3 * log5 8 3

log1/2 75 - log1/2 5^2 log1/2 (75 : 25) log1/2 3

f) .... = --------------------------=--------------------- = ---------- = 1

log1/2 6 - log1/2 2 log1/2 (6 : 2) log1/2 3

log (36 * 4) log 144 log (12 * 12) log 12 + log12 2 * log 12

g) .... = -----------= ----------- = ---------------- = -----------------= ------------ = 2

log (3 * 4) log 12 log 12 log 12 log 12

log 2^2 log 3^2 log 5^2 2 * log 2 2 * log 3 2 * log 5

h) .... = ----------- + ----------- - ------------ = ----------- + ------------ - ------------ =

log 2 log 3 log 5^3 log 2 log 3 3 * log 5

2 + 2 - 2/3 = 4 - 2/3 = 3 i 1/3

Roma

$log _3(log1000) = log_3 3 = 1$

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$log 1000 = x$

$10^x = 1000$

$10^x = 10^3$

$x = 3$

$log_3 3 = x$

$3^x = 3$

$3^x = 3^1$

$x = 1$

$10 \cdot 10 ^{log3-log2} = 10 \cdot 10^{log (3:2)} = 10 \cdot 10^{log 1,5} = 10\ \cdot 1,5 = 15$

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Skorzystano z własności (a, b, c ∈ R+, a ≠ 1 i b ≠ 1):

$log_a b - log_a c = log_a \frac{b}{c}$

$a^{log_a b} = b$

$log _{\frac{1}{2}}(log _39) = log _{\frac{1}{2}}2 = - 1$

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$log_3 9 = x$

$3^x = 9$

$3^x = 3^2$

$x = 2$

$log_{\frac{1}{2}} 2 = x$

$(\frac{1}{2})^x = 2$

$(\frac{1}{2})^x = (\frac{1}{2})^{-1}$

$x = -1$

$10 ^{2+log8} = 10^2 \cdot 10^{log 8} = 100 \cdot 8 = 800$

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Skorzystano z własności (a, b, c ∈ R+, a ≠ 1 i b ≠ 1 oraz m, n ∈ N):

$a^{m + n} =a^m \cdot a^n$

$a^{log_a b} = b$

$\frac{log _536 \cdot log _5 64}{log _56 \cdot log _5 512} = \frac{log _5 6^2 \cdot log _5 2^6}{log _56 \cdot log _5 2^9} = \frac{2 \cdot log _5 6 \cdot 6 \cdot log _5 2}{log _56 \cdot (9 \cdot log _5 2)} = \frac{2 \cdot 6}{9} = \frac{12}{9} = \frac{4}{3} = 1\frac{1}{3}$

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Skorzystano z własności (a, b ∈ R+, r ∈ R):

$log_a b^r = r \cdot \log_a b$

$\frac{log_{\frac{1}{2}}75-2log_{\frac{1}{2}}5}{log_{\frac{1}{2}}6-log_{\frac{1}{2}}2} =\frac{log_{\frac{1}{2}}75-log_{\frac{1}{2}}5^2}{log_{\frac{1}{2}}(6 : 2)} =\frac{log_{\frac{1}{2}}75-log_{\frac{1}{2}}25}{log_{\frac{1}{2}} 3} = \frac{log_{\frac{1}{2}}(75 :25)}{log_{\frac{1}{2}} 3} =$

$= \frac{log_{\frac{1}{2}}3}{log_{\frac{1}{2}} 3} = 1$

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Skorzystano z własności (a, b, c ∈ R+, a ≠ 1 i b ≠ 1, r ∈ R):

$log_a b^r = r \cdot \log_a b$

$log_a b - log_a c = log_a \frac{b}{c}$

$\frac{log36+log4}{log3+log4} = \frac{log(36 \cdot 4)}{log (3 \cdot 4)} = \frac{log144}{log 12} = \frac{log12^2}{log 12} = \frac{2 \cdot log12}{log 12} = 2$