# 503 MrG How do we Solve Logarithmic Equations?

## 2713 days ago by MATH4R2013

#1) Inverse Functions, note log(x) in SAGE is ln(x) #1) f(g(x))==g(f(x)) iff g(x) is the inverse of f(x) #1) f(x) is a reflection of g(x) about y=x iff g(x) is the inverse of f(x) f(x)=e^x g(x)=log(x) show(f(g(x))) show(g(f(x)).simplify()) show(bool(f(g(x))==g(f(x)).simplify())) plot([f(x),g(x),x],0,2,aspect_ratio=1)
 \newcommand{\Bold}{\mathbf{#1}}x \newcommand{\Bold}{\mathbf{#1}}x \newcommand{\Bold}{\mathbf{#1}}\mathrm{True}  #2) graph y=-ln(x+2) using what you know about the graph of y=ln(x) f(x)=ln(x) g(x)=-ln(x) h(x)=-ln(x+2) plot([f(x),g(x)],0,2,aspect_ratio=1)  plot([h(x)],-2,0,aspect_ratio=1)  #3) solve for x: log base 3 (4x-7)==2 solve([log(4*x-7)/log(3)==2],x)
 $\newcommand{\Bold}{\mathbf{#1}}\left[x = 4\right]$
plot([log(4*x-7)/log(3),2],2,5,aspect_ratio=1)  #4) solve for x: log base x 64 == 2 ans=solve([log(64)/log(x)==2],x);show(ans) show(ans) show(ans.rhs()) show(ans.rhs().simplify_radical())
 \newcommand{\Bold}{\mathbf{#1}}\left[x = e^{\left(\frac{1}{2} \, \log\left(64\right)\right)}\right] \newcommand{\Bold}{\mathbf{#1}}x = e^{\left(\frac{1}{2} \, \log\left(64\right)\right)} \newcommand{\Bold}{\mathbf{#1}}e^{\left(\frac{1}{2} \, \log\left(64\right)\right)} \newcommand{\Bold}{\mathbf{#1}}8
plot([log(64)/log(x),2],6,9,aspect_ratio=1)  #5) solve for x: e^(2*x)==5 ans=solve([e^(2*x)==5],x);show(ans) show(ans.rhs().n()) show(ans.rhs().n())
 \newcommand{\Bold}{\mathbf{#1}}\left[x = \log\left(-\sqrt{5}\right), x = \frac{1}{2} \, \log\left(5\right)\right] \newcommand{\Bold}{\mathbf{#1}}0.804718956217050 + 3.14159265358979i \newcommand{\Bold}{\mathbf{#1}}0.804718956217050
plot([e^(2*x),5],0,1)  Note 5a) using ln:

(let ln means log base e)
e^(2*x)==5
ln(e^(2*x))==ln(5)
2*x==ln(5)
x==ln(5)/2

Note 5b)using log:

(let log means log base 10)
e^(2*x)==5
log(e^(2*x))==log(5)
2*x*log(e)==log(5)
x==1/2*log(5)/log(e)

Note 5c) change of base formula (from unknown base to known base logs)!

therefore, log base b of a = (known log of a)/(known log of b)