Use the properties of logarithms to write the expressions in...

Use the properties of logarithms to write the expressions in Exercises.
a. $\ln \sin \theta-\ln \left(\frac{\sin \theta}{5}\right)$
b. $\ln \left(3 x^{2}-9 x\right)+\ln \left(\frac{1}{3 x}\right)$
c. $\frac{1}{2} \ln \left(4 t^{4}\right)-\ln b$

Key Concepts

Logarithm Function

The logarithm function is the inverse of an exponential function. It is defined for positive real numbers, and its properties allow us to convert multiplication into addition and vice versa, making it a powerful tool for simplifying expressions, solving equations, and analyzing growth or decay processes.

Product Rule

The product rule for logarithms states that the sum of two logarithms with the same base can be rewritten as the logarithm of the product of the two arguments, i.e., ln(a) + ln(b) = ln(ab). This rule enables simplification of expressions where separate logarithmic terms are added.

Quotient Rule

The quotient rule for logarithms expresses that the difference between two logarithms with the same base is equal to the logarithm of the quotient of the arguments, i.e., ln(a) - ln(b) = ln(a/b). This rule is particularly useful for condensing logarithmic expressions where one logarithm is subtracted from another.

Power Rule

The power rule for logarithms allows a constant multiplier in front of a logarithm to be expressed as an exponent inside the logarithm, namely k·ln(a) = ln(a^k). This rule is essential for simplifying expressions where logarithms involve coefficients, as it transforms the expression into a more tractable form.

Transcript

00:02 Today i'm going to practice simplifying some logarithmic expressions using the properties of logs.

00:08 So for example, if we have the natural log of sine of theta minus the natural log of sine of theta over five, we can then actually figure out what this simplifies to.

00:28 So because we have a subtraction sign, we know we can rewrite this as the natural log of sine.

00:37 Here being our numerator over this whole thing here as our denominator, sine of theta over five.

00:53 And when you get a fraction over a fraction, we have this main fraction here and another fraction down here.

00:59 We can flip the bottom fraction and multiply on the inside here.

01:04 So we get sine of theta times five over sine of theta, and therefore the signs cancel.

01:13 So the natural log of 5 would be our solution.

01:18 And this, of course, assumes that sine theta is on the appropriate domain, that it's greater than 0.

01:24 So, let's look at another example.

01:26 Let's say we do the natural log of, how about 3x squared minus 9x plus the natural log of 1 over 3x.

01:42 Because we have a plus sign here, we can multiply the quantities inside the natural logs.

01:47 So we get 3x squared minus 9x times 1 over 3x.

01:57 And so we can simplify that and say the natural log of 3x squared over 3x minus 9x over 3x.

02:09 Simplifying what's inside 3x squared divide by 3x, 3 is cancel, the x is canceled down 1, leaving just x...

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