Question

solve for y to 4 decimal places in; log base y for 10+log base 10 for y = 5

Answer 1

\[\log_y(10)+\log_{10}(y)=5\]
?

Answer 2

let's try this. by the change of base formula
\[\log_y(10)=\frac{\log_{10}(10)}{\log_{10}(y)}=\frac{1}{log_{10}(y)}\]\]

Answer 3

giving
\[\frac{1}{\log(y)}+\log(y)=5\] and if we put
\[\log(y)=x\] we have
\[\frac{1}{x}+x=5\]
\[1+x^2=5x\]
\[x^2-5x+1=0\]
\[x=\frac{5\pm\sqrt{21}}{2}\]

Answer 4

now
\[\log(y) = \frac{5+\sqrt{21}}{2}\]
\[y=10^{\frac{5+\sqrt{21}}{2}}\] whatever that is. guess you need a calculator for that