91Ó°ÊÓ

Converting between exponential forms and logarithmic forms is essential when solving exponential equations. In exponential format, the equation is often structured as \( b^x = y \), where \( b \) is the base, \( x \) is the exponent, and \( y \) is the result.

To convert this to logarithmic form, we use the statement \( \,\log_b y = x \, \). Here, the base \( b \) remains the same, \( y \) becomes the value you take the logarithm of, and \( x \) is what the logarithm equals. This conversion is helpful as it allows us to solve for the exponent \( x \) using logarithms.

In the given problem, after isolating the term with \( x \), we had \( 3^x = 4.2 \). By converting this to logarithmic form, it became \( \,\log_3{4.2} = x \, \). This tells us that \( x \) is what the logarithm to the base 3 of 4.2 equals. This conversion simplifies the process of finding the unknown exponent.

Solving for variables often involves isolating the desired variable on one side of the equation. This requires removing other elements or terms that may interfere with isolating the variable.

In this case, the variable \( x \) was embedded within the exponential equation \( 3^x + 0.7 = 4.9 \). To isolate \( x \), we first subtracted 0.7 from both sides to have \( 3^x = 4.2 \). The isolation of the base power term is crucial for simplifying the problem and making logarithmic conversion possible.

Once the base power term is isolated, you can convert it to logarithmic form to solve for \( x \). This step is important as it directly leads to finding the exponent value that satisfies the given equation.

Rounding numbers is a fundamental mathematical skill that ensures results are presented in a manageable form, often required in practical problem-solving.

When dealing with logarithms, calculators typically provide results with many decimal places. It's common to round such results, especially in practical applications, to keep them easier to interpret and report. In our solution step, calculating \( \,\log_3{4.2} \, \) using a calculator likely yielded a long decimal.

The problem specified that results should be rounded to the nearest ten-thousandth. This means you'd look at the fifth decimal place and determine whether to round up or leave the nearest ten-thousandth digit as it is.

• If the fifth decimal is 5 or higher, round the fourth decimal place up.
• If it's lower than 5, keep the fourth decimal place unchanged.

These steps ensure that the result maintains the precision necessary for the context of the problem while remaining practical and presentable.