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How do you solve $4\left( {{7}^{x+2}} \right)={{9}^{2x-3}}?$

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Last updated date: 01st Mar 2024
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IVSAT 2024
Answer
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Hint: Use natural $\log $ as normal $\log .$ in the given equation.
Remember the rule of logarithm.
$\log a.b=\log +\log b$ and $\log {{x}^{a}}=a\log x$
A logarithmic model is a model that measures the magnitude of the things it's measuring. It can also be seen as the inverse of an exponential model.

Complete step by step solution:
As per the question the given equation is
$4\left( {{7}^{x+2}} \right)={{9}^{2x-3}}?$
Here, use either natural $\log $ or normal $\log $. In or $\log $ and $\log $ on both sides.
Therefore the modified equation will be,
\[\ln \left( {{4.7}^{x+2}} \right)=\ln \left( {{9}^{2x-3}} \right)\]
Here, we can use logarithmic rules.
$\log a.b=\log a+\log b$ in the $\ln \left( {{4.7}^{x+2}} \right)$ term.
$\ln \left( 4 \right)+\ln \left( {{7}^{x+2}} \right)=\ln \left( {{9}^{2x-3}} \right)$
Remember the rule of logarithm that states
$\log {{x}^{4}}=4\log x$
$\ln \left( 4 \right)+\left( x+2 \right)\ln \left( 7 \right)=\left( 2x-3 \right)\ln \left( 9 \right)$
Bring all the $'x'$ terms one side and $'\ln '$ terms one side.
$x\ln \left( 7 \right)-2x\ln \left( 9 \right)=-3\ln \left( 9 \right)-2\left( 7 \right)-\ln \left( 4 \right)$
Factorise the $'x'$ out.
$x\left( \ln \left( 7 \right)-2\ln \left( 9 \right) \right)=\left( -3\ln \left( 9 \right)-2\ln \left( 7 \right)-\ln \left( 4 \right) \right)$
$x=\dfrac{-3\ln \left( 9 \right)-2\ln \left( 7 \right)-\ln \left( 4 \right)}{\ln \left( 7 \right)-2\ln \left( 9 \right)}$
Side on the calculator using the $\ln $ button or if your calculator doesn’t have it use the $\log $ base $10$ button.
So,
$x=\dfrac{-11.86}{-2.44}$
$\left( - \right)$ of numerator and denominator gets canceled.
$x=\dfrac{11.86}{2.44}$
Hence,
$x=4.86$

Additional Information:
There are two main advantages of logarithms model:
(1) Linearization
(2) Ease of computational cooperation, the farmer of which ties into the second, The easier one to explain is the ease of computational cooperation. The logarithmic system thinks that simple to explain is the pH model, which most people are at least vaguely geurae, you see the ph is actually a mathematical code for ‘minus’ log of’ so pH is actually $-\log \left( H \right)$
Exponential growth is key common in nature for things like radioactivity, bacterial growth etc. being written as,
$N\left( t \right)={{N}_{0}}{{e}^{Kt}}$ or $N\left( t \right)={{N}_{0}}{{a}^{t}}$
So, if we wanted to know how much time passed by based on the amount there is, We would have a logarithmic model.
$\dfrac{\ln \left[ \dfrac{N\left( t \right)}{{{N}_{0}}} \right]}{K}=t$or ${{\log }_{a}}\left[ \dfrac{N\left( t \right)}{{{N}_{0}}} \right]=t$

Note: Use the natural $\log $ or normal $\log $ or $\log $ on both sides.
Also apply the logarithmic rule,
$\log a.b=\log a+\log b$
Use the logarithmic rule.
i.e. $\log {{x}^{a}}=a\log x$
and at least to find the value of $'x'$ you have to separate $'x'$ from $\ln $ term and after that you can also put the value in the calculator to find the value of $'x'$
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