Changing bases Convert the following expressions to the indicated base.

$\ln ∣ x ∣$ using base 5

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insert step 1: Recall the change of base formula for logarithms, which is $ \log_b a = \frac{\log_k a}{\log_k b} $, where $ b $ is the new base and $ k $ is the current base.

insert step 2: Identify the current expression $ \ln|x| $, which is a natural logarithm with base $ e $.

insert step 3: Apply the change of base formula to convert $ \ln|x| $ to base 5: $ \log_5|x| = \frac{\ln|x|}{\ln 5} $.

insert step 4: Recognize that $ \ln|x| $ is the numerator and $ \ln 5 $ is the denominator in the change of base formula.

insert step 5: The expression $ \ln|x| $ in base 5 is $ \frac{\ln|x|}{\ln 5} $.

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주요 개념

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Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another. It states that for any positive numbers a, b, and x (where a and b are not equal to 1), the logarithm can be expressed as log_b(x) = log_a(x) / log_a(b). This is essential for solving logarithmic expressions in different bases.

Natural Logarithm

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and mathematical analysis due to its unique properties, such as the derivative of ln(x) being 1/x. Understanding natural logarithms is crucial when converting expressions involving ln to other bases.

Absolute Value in Logarithms

The absolute value in logarithmic expressions, such as ln|x|, indicates that the logarithm is defined for both positive and negative values of x, as long as x is not zero. This is important because logarithms are only defined for positive arguments, and the absolute value ensures that the input remains valid regardless of the sign of x.

Changing bases Convert the following expressions to the indicated base.ln∣x∣\(\ln\]\left\)|x\(\right\)| using base 5