Question

Rewrite each expression in an equivalent form that only contains one base 10 logarithm.

Answer 1

Answer:

$-\frac{1}{\log(x)}$

Step-by-step explanation:

Data provided:

logₓ (1/10)

Now,

From the properties of log function

$\log\frac{A}{B}$ = log(A) - log(B)

logₓ (z)= $\frac{\log(z)}{\log(x)}$  (where the base of the log is equal for both numerator and the denominator)

also,

log(xⁿ) = n × log(x)

Therefore,

logₓ (1/10) = logₓ(1) - logₓ(10)

=  $\frac{\log(1)}{\log(x)} - \frac{\log(10)}{\log(x)}$  (here log = log₁₀)

=  $\frac{0}{\log(x)} - \frac{\log(10)}{\log(x)}$ (as log(1) = 0)

=   $-\frac{\log(10)}{\log(x)}$

=  $-\frac{1}{\log(x)}$     (as log(10) = 1)

Therefore,

the given equation can be rewritten as  $-\frac{1}{\log(x)}$