if \(\log_{10}(x) = 3 + \log_{10}(y)\), find \(\dfrac{x}{y}\).

Let x and b be positive real numbers so that \(\log_b(x^2) = 10\) Find \(\log_{\sqrt[3]{b}} \left( \frac{1}{x} \right)\)

so i've been stuck on these problems.... im pretty sure we apply some log formula to it but im not sure what.

help appreciated!

thank you!!

Guest Mar 28, 2022

#1

#2**+2 **

I *think* this is how to do this one:

log_{b} (x^2) = 10

2 log_{b }(x) = 10

log_{b} (x) = 5

Now use the log base-change rule for the second part ( to change to base 'b' )

= log_{b} (1/x) / log_{b} ( b^{1/3})

= log_{b }(x^{-1}) / ( 1/3 log_{b}_{ }(b) ) remember log_{b }(b) = 1

= (-1) log_{b} (x) / (1/3)

= -3 log_{b}(x) = - 3 ( 5) = -15

ElectricPavlov
Mar 29, 2022