Ambiguity with Logs and Inequalities Solve the following inequality: 0.8^x>0.4

Amy Bright

Amy Bright

Answered question

2022-11-19

Ambiguity with Logs and Inequalities
Solve the following inequality:
0.8 x > 0.4
Method 1 (Using the Common Logarithm)
log 10 0.8 x > log 10 0.4
x log 10 0.8 > log 10 0.4
Because
log 10 0.8 < 0
The sign of the inequality will change when we divide both sides of the equation by the LHS of the above inequality:
x < log 10 0.4 log 10 0.8
Using a reverse of the change of base theorem, we get:
x < log 0.8 0.4
Method 2 (Using another based Logarithm)
log 0.8 0.8 x > log 0.8 0.4
x log 0.8 0.8 > log 0.8 0.4
Because
log 0.8 0.8 > 0
The sign of the inequality will not change when we divide both sides of the equation by the LHS of the above inequality:
x > log 0.8 0.4 log 0.8 0.8
The denominator of the RHS of the inequality is equal to one, therefore:
x > log 0.8 0.4
The sign is reversed using both methods - the textbook states the answer to be the one gotten using the first method, but I can't spot where I went wrong in method two.
Any help will be greatly appreciated, thanks in advance.

Answer & Explanation

Aiden Villa

Aiden Villa

Beginner2022-11-20Added 10 answers

The method 2 is wrong at the very beginning because the following is wrong :
0.8 x > 0.4 log 0.8 0.8 x > log 0.8 0.4
Instead we have
0.8 x > 0.4 log 0.8 0.8 x < log 0.8 0.4
because 0.8 < 1
For a > 1
b > c > 0 log a b > log a c .
For a < 1
b > c > 0 log a b < log a c .

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