Math Simplification: Breaking Down an Expression Using Logarithmic Properties

(a^u)(1/ln . a) + c

To work on this expression, we will first use the properties of logarithms to simplify it

To work on this expression, we will first use the properties of logarithms to simplify it.

Given the expression:

(a^u)(1/ln a) + c

Let’s break it down step-by-step:

Step 1: Simplify (1/ln a)

The expression (1/ln a) can be simplified using the properties of logarithms. ln a represents the natural logarithm of a.

Recall that the natural logarithm, ln x, is the logarithm to the base e, where e is a mathematical constant approximately equal to 2.71828.

Using the property of logarithms, 1/ln a is equivalent to ln(a)^(-1):

1/ln a = ln(a)^(-1)

Step 2: Rewrite the expression with the simplified term

Substituting ln(a)^(-1) back into the original expression, we get:

(a^u)(ln(a)^(-1)) + c

Step 3: Combine the terms

In order to combine the terms (a^u) and (ln(a)^(-1)), we can use the properties of exponents and logarithms.

Recall that the product of two exponential terms with the same base can be combined by adding their exponents. Similarly, the division of two exponential terms with the same base can be combined by subtracting their exponents.

In this case, we have (a^u) and (ln(a)^(-1)), where the base is different. To combine these terms, we need to find a common base. Since (ln a) is the natural logarithm, we’ll choose e as the common base.

Using the change of base formula, we rewrite (ln(a))^(-1) as (e^(-1))/(ln e):

(a^u)(ln(a)^(-1)) + c = (a^u)((e^(-1))/(ln e)) + c

Step 4: Simplify further if needed

At this point, you should check if there are any further simplifications or instructions given in the problem, or if any additional specific values are given for a, u, or c.

If no further information is provided, this is the simplified form of the given expression:

(a^u)(ln(a)^(-1)) + c = (a^u)((e^(-1))/(ln e)) + c

Note: It is important to use parentheses to clearly indicate the order of operations in the expression.

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