91Ó°ÊÓ

Natural logarithms (\text{ln}) are logarithms with a base of the mathematical constant \(e\), which is approximately 2.718. Natural logarithms are commonly used in solving exponential equations due to their properties that make it easier to handle complex expressions.
To solve an exponential equation like \(4^{x-2}=5^{3x+2}\), taking the natural logarithm on both sides helps to linearize the equation, allowing us to work with exponents in a more straightforward manner.
Using natural logarithms simplifies many steps in solving these problems by transforming multiplication into addition and handling large exponents efficiently.

The power rule for logarithms states that for any positive number \(a\) and any real number \(b\), the logarithm of \(a^b\) is equal to \(b\text{ln}(a)\). Mathematically, this is expressed as \(\text{ln}(a^b)=b\cdot\text{ln}(a)\).
In our problem, applying the power rule allows us to bring down the exponents so we can work with linear equations rather than exponential ones.
For example:

• Starting with \(\text{ln}(4^{x-2})\), applying the power rule transforms it to \((x-2)\cdot\text{ln}(4)\).
• Similarly, \(\text{ln}(5^{3x+2})\) becomes \((3x+2)\cdot\text{ln}(5)\).

Once the exponents are brought down, solving the equation becomes much simpler.

Isolating variables is a critical step in solving algebraic equations. It involves moving terms with the variable of interest to one side of the equation and constants to the other side.
In our scenario, after applying the power rule, we get a linear equation: \((x-2)\text{ln}(4) = (3x+2)\text{ln}(5)\).
We first distribute the logarithms to get: \(x\text{ln}(4) - 2\text{ln}(4) = 3x\text{ln}(5) + 2\text{ln}(5)\).
Then, we combine like terms to isolate \(x\): \(x\text{ln}(4) - 3x\text{ln}(5) = 2\text{ln}(4) + 2\text{ln}(5)\).
Finally, factoring out \(x\) on the left side gives us: \(x (\text{ln}(4) - 3\text{ln}(5)) = 2\text{ln}(4) + 2\text{ln}(5)\).

When solving equations, the exact solution might not always be practical or possible to obtain. In such cases, we approximate the solution to a certain number of decimal places.
For our problem, the final step requires calculating an exact numerical value of \(x\). For that, we approximate the natural logarithms to three decimal places:

• \(\text{ln}(4)\) is approximately 1.386.
• \(\text{ln}(5)\) is approximately 1.609.

Substituting these values into the equation \( x = \frac{2\text{ln}(4) + 2\text{ln}(5)}{\text{ln}(4) - 3\text{ln}(5)} \), we get:
\( x \frac{2(1.386) + 2(1.609)}{1.386 - 3(1.609)} \approx \frac{5.99}{-3.44} \approx -1.742.\) This gives our approximate solution: \(x \approx -1.742.\). Approximations let us understand the scale of the answer while keeping calculations manageable.