Find the result of log 10(7)*log 10(343)*log 7(100)* log 7( 1000).

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We'll write conventionally the decimal logarithms as log 10 x = lg x

We'll try to write the arguments of logarithms, except the prime numbers, as powers.

For instance:

343 = 7*7*7 = 7^3

We'll take logarithms both sides and we'll apply product rule:

lg 343 = lg(7*7*7) = lg7 + lg7 + lg7 = 3lg7

100 = 10*10 = 10^2

We'll take logarithms both sides and we'll apply product rule:

log 7 100 = log 7 (10*10) = log 7 10 + log 7 10 = 2 log 7 10

1000 = 10*10*10 = 10^3

We'll take logarithms both sides and we'll apply product rule:

log 7 1000 = log 7 (10*10*10) = log 7 10+ log 7 10 + log 7 10 = 3 log 7 10

The expression will become:

lg(7)*(3lg7)*(2 log 7 10)* (3 log 7 10) = 18[(lg 7)^2]*(log 7 10)^2

But log 7 10 = 1/lg 7

18[(lg 7)^2]*(log 7 10)^2 = 18[(lg 7)^2]/(lg 7)^2

We'll simplify and we'll get:

18[(lg 7)^2]*(log 7 10)^2 = 18

The expression `log_10 7*log_10 343*log_7 100* log_7 1000` has to be simplified.

Use the following properties of logarithm` log_b a = 1/(log_a b)` and `log a^b = b*log a`

`log_10 7*log_10 343*log_7 100* log_7 1000`

= `log_10 7*log_10 7^3*log_7 10^2* log_7 10^3`

= `log_10 7*3*log_10 7*2*log_7 10*3*log_7 10`

= `1/(log_7 10)*3*log_10 7*2*log_7 10*3*1/(log_10 7)`

= 3*2*3

= 18

The value of `log_10 7*log_10 343*log_7 100* log_7 1000 = 18`

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