Solve the exponential equation $2^{\left(x-1\right)}=3^{\left(2x-1\right)}$

Step-by-step Solution

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Final answer to the problem

$x=\frac{\ln\left(\frac{2}{3}\right)}{\ln\left(2\right)-\ln\left(9\right)}$
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Apply the formula: $y=x$$\to \ln\left(y\right)=\ln\left(x\right)$, where $x=3^{\left(2x-1\right)}$ and $y=2^{\left(x-1\right)}$

$\ln\left(2\right)\left(x-1\right)=\ln\left(3\right)\left(2x-1\right)$

Learn how to solve equations exponentielles problems step by step online.

$\ln\left(2\right)\left(x-1\right)=\ln\left(3\right)\left(2x-1\right)$

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Learn how to solve equations exponentielles problems step by step online. Solve the exponential equation 2^(x-1)=3^(2x-1). Apply the formula: y=x\to \ln\left(y\right)=\ln\left(x\right), where x=3^{\left(2x-1\right)} and y=2^{\left(x-1\right)}. Apply the formula: x\left(a+b\right)=xa+xb, where a=x, b=-1, x=\ln\left(2\right) and a+b=x-1. Apply the formula: x\left(a+b\right)=xa+xb, where a=2x, b=-1, x=\ln\left(3\right) and a+b=2x-1. Apply the formula: a\ln\left(x\right)=\ln\left(x^a\right).

Final answer to the problem

$x=\frac{\ln\left(\frac{2}{3}\right)}{\ln\left(2\right)-\ln\left(9\right)}$

Exact Numeric Answer

$x=0.2695773$

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Function Plot

Plotting: $2^{\left(x-1\right)}- 3^{\left(2x-1\right)}$

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9
0
a
b
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d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
2

e
π
ln
log
log
lim
d/dx
Dx
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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