Solution - Exponential equations using logarithms

x = \log_{6} (42)

x=log_6(42)

Decimal form:

x = 2.086033132501692

x=2.086033132501692

See steps

Step-by-step explanation

1. Remove the variable from the exponent using logarithms

$6^{x} \geq 42$

Take the common logarithm of both sides of the equation:

$\log_{10} (6^{x} >) = \log_{10} (42)$

Use the log rule: $\log_{a} (x^{y}) = y \cdot \log_{a} (x)$ to move the exponent outside the logarithm:

$x \cdot \log_{10} (6) = \log_{10} (42)$

2. Isolate the x-variable

$x \cdot \log_{10} (6) = \log_{10} (42)$

Divide both sides of the equation by $\log_{10} (6)$ :

$x = \frac{l o g_{10} (42)}{l o g_{10} (6)}$

Use the formula $\frac{l o g_{b} (x)}{l o g_{b} (a)} = l o g_{a} (x)$ to combine the logarithms into one:

$x = \log_{6} (42)$

Decimal form:

$x = 2.086033132501692$

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Why learn this

Exponential functions are used to represent the data of the rapid growth and decay of materials, proportional to their current amount. There are many natural processes that can be represented using exponential math models, including radioactive decay, atmospheric pressure change accompanying altitude change (for example, an ascending or descending airplane), bacterial growth, population growth, and the spread of viruses. Therefore, understanding exponential functions will allow you to better interpret data and put you one step closer to a career in a number of interesting fields, such as finance, medicine, aeronautics, and many others.

Terms and topics

Exponential equations