This is the equation that I am having troubles with: x^(log_(10)5)+5^(log_(10)x)=50 So the first thing I do, I logarithm the whole expression with log_(10). So I get: log_(10)5xxlog_(10)x+log_(10)5xxlog_(10)x=log_(10)50 When I solve this one for x, I get that x=16, which is totally incorrect because it is supposed to be 100. Can anyone tell me what am I doing wrong or show me how to solve this equation?

jhenezhubby01ff

jhenezhubby01ff

Answered question

2022-09-29

Solving exponential equations using logarithms
This is the equation that I am having troubles with:
x log 10 5 + 5 log 10 x = 50
So the first thing I do, I logarithm the whole expression with log 10
So I get:
log 10 5 × log 10 x + log 10 5 × log 10 x = log 10 50
When I solve this one for x, I get that x = 16, which is totally incorrect because it is supposed to be 100. Can anyone tell me what am I doing wrong or show me how to solve this equation?

Answer & Explanation

Rachael Singh

Rachael Singh

Beginner2022-09-30Added 4 answers

Let y = x log 10 5 , then
log 10 y = ( log 10 5 ) ( log 10 x ) = log 10 5 log 10 x y = 5 log 10 x .
Hence
x log 10 5 + 5 log 10 x = 50 5 log 10 x + 5 log 10 x = 50 2 × 5 log 10 x = 50 5 log 10 x = 25 5 log 10 x = 5 2 log 10 x = 2 x = 10 2 = 100 .
sailorlyts14eh

sailorlyts14eh

Beginner2022-10-01Added 1 answers

Your first action was bad, because the identity you assumed (" log a + b = log a + log b ") simply doesn't exist.

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