Question

Simplify each expression using the properties of natural logs.​

Answer 1

To simplify an expression using properties of natural logs, apply the rules of logarithms, which include converting log products into sums, powers into multiplications, and using ℓn and e as inverse functions to simplify equations. Calculators can be used to find natural logarithms or calculate values from them.

Step-by-step explanation:

To simplify an expression using the properties of natural logs, you can apply a series of rules that govern logarithmic functions. The natural logarithm function, denoted as ℓn, has properties that are particularly useful for simplifying expressions involving exponential and logarithmic forms.

• Take the natural logarithm of both sides to cancel the exponential function, for example, if you have an equation of the form e^x = b, applying ℓn to both sides gives ℓn(e^x) = ℓn(b), and since ℓn and e are inverse functions, this simplifies to x = ℓn(b).
• The logarithm of a product of two numbers is the sum of the logarithms of the two numbers, i.e. ℓn(xy) = ℓn(x) + ℓn(y).
• The logarithm of a number raised to an exponent is the product of the exponent and the logarithm of the number, so ℓn(x^y) = y·ℓn(x).
• To find the natural logarithm of a number, you can use the ℓn button on a calculator. To calculate a number from its natural logarithm, you take the inverse ℓn of the natural logarithm, or calculate e^x where x is the natural logarithm of the number.

For instance, to find the natural logarithm of 5.6/16.0, you would use the property that ℓn(a/b) = ℓn(a) - ℓn(b), which gives ℓn(5.6) - ℓn(16.0), and using a calculator, this is approximately -1.050. Using logarithms as exponents facilitates these operations, understanding that operations involving logarithms follow the same rules as operations involving exponents.

Answer 2

`\begin{array}{llll} \textit{Logarithm Cancellation Rules} \\\\ log_a a^x = x\qquad \qquad a^(log_a x)=x \end{array} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \ln\left( e^(-6x) \right)\implies \log_e\left( e^(-6x) \right)\implies -6x \\\\\\ e^(4\ln(x))\implies e^(\ln(x)4)\implies e^(\log_e(x)4)\implies \left[ e^(\log_e(x)) \right]^4\implies [x]^4\implies x^4 \\\\\\ e^(\ln(4x)+2)\implies e^(\ln(x))\cdot e^2\implies e^(\log_e(x))\cdot e^2\implies xe^2`