How do you solve #log_3z=4log_z3#?

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Answer 1 · 2016-01-29T16:10:47

#z=9#

Rewrite everything using the change of base formula.

The change of base formula provides a way of rewriting a logarithm in terms of another base, like follows:

#log_ab=log_cb/log_ca#

In this case, the new base I will choose is #e#, so we will use the natural logarithm.

#log_3z=4log_z3#

#=>lnz/ln3=(4ln3)/lnz#

Cross multiply.

#=>(lnz)^2=4(ln3)^2#

Take the square root of both sides.

#=>lnz=2ln3#

We can modify the right hand side using the rule: #b*lna=ln(a^b)#

#=>lnz=ln(3^2)=ln9#

Use the fairly intuitive rule that if #lna=lnb#, then #a=b#.

#=>z=9#