Question

Solve the equation.\(11^{6 x}=38\)

Short answer

The solution to the equation \(11^{6 x}=38\) is \(x \approx 0.3987\).

Step-by-step solution

Apply logarithm to both sides

First, apply the natural logarithm to both sides. This results in: \(ln (11^{6 x}) = ln (38)\). Using the logarithm rule \(ln (a^b) = b* ln a\), this simplifies to: \(6x*ln (11) = ln (38)\).

Isolate x

Now, isolate x to one side of the equation, giving: \( x = \frac{ln(38)}{6*\ln(11)}\).

Compute the value

Perform the division to yield the numerical value of x which is approximately: 0.3987.

Key concepts

Natural Logarithm

When solving exponential equations, the natural logarithm, denoted as \( \text{ln} \), plays a critical role. The natural logarithm is the inverse function of the exponential function \( e^x \), where \( e \) is an important mathematical constant approximately equal to 2.71828.

In practical terms, \( \text{ln}(x) \) answers the question: What power must \( e \) be raised to in order to yield \( x \)? For example, since \( e^2 \thtrightta\roximately 7.389\), we say \( \text{ln}(7.389) \thtrightta\roximately 2 \). In this process, we effectively 'bring down' the exponent in an equation, making it much easier to solve for the unknown variable.

Isolating Variables

To solve for an unknown in any equation, one must isolate the variable of interest. Isolating variables is a fundamental technique not just in algebra, but in all areas of mathematics.

The process involves using algebraic operations to get the variable on one side of the equation and everything else on the other side. Here, by applying the natural logarithm, we've simplified the original exponential equation to a form that allows for easier manipulation. By subsequently dividing both sides by \( 6\text{ln}(11) \), we isolate \( x \), and thus, pave the way to finding its value. Understanding how to juggle mathematical operations to isolate the variable is the key to unlocking solutions to numerous mathematical problems.

Logarithmic Properties

Logarithmic properties are rules that make working with logarithms easier and allow us to solve exponential equations. Three fundamental properties are frequently used: the product property \( \text{ln}(ab) = \text{ln}(a) + \text{ln}(b) \), the quotient property \( \text{ln}\tleft(\frac{a}{b}\right) = \text{ln}(a) - \text{ln}(b) \), and the power property \( \text{ln}(a^b) = b \text{ln}(a) \).

In the given solution, the power property is used to bring down the exponent, changing \( \text{ln}(11^{6x}) \) to \( 6x \text{ln}(11) \). These properties ultimately transform complex exponential forms into more manageable linear ones, which can then be manipulated with basic algebraic techniques to reveal the value of the unknown variable.