Answer
(a) $x = 25$
(b) $x=2$
(c) $x=11.13$
(d) $x=5.39$
Work Step by Step
A little review:
$a^b=a^x$ => $b=x$
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(a)
$3^{4x}=3^{100}$
$4x=100$
$x=25$
(b)
$e^{3x-2}=e^{x^2}$
$3x-2=x^2$
$x^2-3x+2=0$
$x_1=1$; $x_2=2$
(c)
$5^{\frac{x}{10}}+1=7$
$5^{\frac{x}{10}}=6$ //Then we apply logarithm rule
$\log_56=\frac{x}{10}$
$x=10\log_56\approx11.13$
(d)
$10^{x+3}=6^{2x}$ //Let's take $\log_{10}$ from the both side
$\log10^{x+3}=\log6^{2x}$
$(x+3)\log10=(2x)\log6$ //Note, omitting base means to have base $10$
$x+3=(2x)\log6$
$x=2x\log6-3$
$x-2x\log6=-3$
$x(1-2\log6)=-3$
$x=\frac{3}{2\log6-1}\approx5.39$
