Question

A teacher used the change of base formula to determine whether the equation below is correct.

(log Subscript 2 Baseline 10) (log Subscript 4 Baseline 8) (log Subscript 10 Baseline 4) = 3


Which statement explains whether the equation is correct?

Answer 1

B on E2020.

Explanation:

Answer 2

The equation is correct

Explanation:

The equation, written as:

[log_2 (10)][log_4 (8)][log_10 (4)] = 3

Consider the change of base formula:

log_a (x) = [log_10 (x)]/ [log_10 (a)]

Applying the change of base formula to change the expressions in base 2 and base 4 to base 10.

(1)

log_2 (10) = [log_10 (10)]/[log_10 (2)]

= 1/[log_10 (2)]

(Because log_10 (10) = 1)

(2)

log_4 (8) = [log_10 (8)]/[log_10 (4)]

Now putting the values of these new logs in base 10 into the left-hand side of original equation to verify if we have 3, we have:

[log_10 (2)][log_8 (4)][log_10 (4)]

= [1/ log_10 (2)][log_10 (8) / log_10 (4)][log_10 (4)]

= [1/log_10 (2)] [log_10 (8)]

= [log_10 (8)]/[log_10 (2)]

= [log_10 (2³)]/[log_10 (2)]

Since log_b (a^x) = xlog_b (a)

= 3[log_10 (2)]/[log_10 (2)]

= 3 as required

Therefore, the left hand side of the equation is equal to the right hand side of the equation.