Objective 4 - Solve Exponential - Ximera

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Solve exponential equations with the same or different bases.

Link to section in online textbook.

First, watch this video to learn how to solve certain types of exponential equations (when the bases are the same). Then practice this method below.

Solve the exponential equation below.

$\sage {base5}^{\sage {factor5a}} = \sage {base5}^{\sage {factor5b}}$

$x = \answer [tolerance=0.05]{\sage {solution5}}$

Solve the exponential equation below.

$\sage {base6a}^{\sage {factor6a}} = \sage {base6b}^{\sage {factor6b}}$

$x = \answer [tolerance=0.05]{\sage {solution6}}$

Solve the exponential equation below.

$\left (\sage {base7a}\right )^{\sage {factor7a}} = \sage {base7b}^{\sage {factor7b}}$

$x = \answer [tolerance=0.05]{\sage {solution7}}$

Solve the exponential equation below.

$\sage {base8a}^{\sage {factor8a}} = \sage {base8b}^{\sage {factor8b}}$

$x = \answer [tolerance=0.05]{\sage {solution8}}$

It’s great when equations line up like this, but what happens if we can’t write the bases to be the same? Watch this video to learn how to solve any type of exponential equation. Then practice with the questions below.

Note: Yes, you can use this method all of the time! While changing the bases to be the same to rewrite the question can be easier sometimes, taking the log of both sides will always solve exponential equations.

Solve the exponential equation below.

$\sage {base9a}^{\sage {factor9a}} = \sage {base9b}^{\sage {factor9b}}$

$x = \answer [tolerance=0.05]{\sage {solution9}}$

Solve the exponential equation below.

$\sage {base10a}^{\sage {factor10a}} = \sage {base10b}^{\sage {factor10b}}$

$x = \answer [tolerance=0.05]{\sage {solution10}}$

Solve the exponential equation below.

$\sage {base11a}^{\sage {factor11a}} = \sage {base11b}^{\sage {factor11b}}$

$x = \answer [tolerance=0.05]{\sage {solution11}}$

Main takeaway: Before looking, you should work through the previous problems. Have you finished working through the examples? $\answer [format=string]{Yes}$

While changing the bases to be the same to rewrite the question can be easier sometimes, taking the log of both sides will always solve exponential equations.

Start typing the name of a mathematical function to automatically insert it.

(For example, "sqrt" for root, "mat" for matrix, or "defi" for definite integral.)

Symbols

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

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