Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$(iv) f(x)=(x+(1)/(2))log_((x+(1)/(2))){(x^(2)+2x-3)/(4x^(2)-4x-3)}$

(iv) $f(x)=\left(x+\frac{1}{2}\right) \log _{\left(x+\frac{1}{2}\right)}\left\{\frac{x^{2}+2 x-3}{4 x^{2}-4 x-3}\right\}$

View step-by-step help

View step-by-step help

Quotient property of logarithms

Full solution

Q. (iv)

f(x)=\left(x+\frac{1}{2}\right) \log _{\left(x+\frac{1}{2}\right)}\left\{\frac{x^{2}+2 x-3}{4 x^{2}-4 x-3}\right\}

Identify Base and Argument: Let's first identify the base of the logarithm and the argument of the logarithm in the function $f(x)$ . The base of the logarithm is $(x + \frac{1}{2})$ , and the argument is the fraction $\frac{x^2 + 2x - 3}{4x^2 - 4x - 3}$ .
Apply Change of Base Formula: We can apply the change of base formula to the logarithm to convert it to a common logarithm (base $10$ ) or natural logarithm (base $e$ ). The change of base formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$ , where $c$ is the new base. However, since the base of the logarithm and the coefficient in front of the logarithm are the same, the logarithm simplifies to $1$ when evaluated at the base, which means $f(x)$ simplifies to the argument of the logarithm.
Simplify $f(x)$ to Argument: Therefore, we can simplify $f(x)$ to just the argument of the logarithm, which is $\frac{x^2 + 2x - 3}{4x^2 - 4x - 3}$ , since the logarithm of a number at its own base is $1$ .
$f(x) = (x + \frac{1}{2}) \cdot 1 \cdot \left(\frac{x^2 + 2x - 3}{4x^2 - 4x - 3}\right)$
Multiply by Fraction: Now, we can simplify the expression by multiplying $(x + \frac{1}{2})$ by the fraction. $\newline$ $f(x) = (x + \frac{1}{2}) \times \left(\frac{x^2 + 2x - 3}{4x^2 - 4x - 3}\right)$
Correct Conceptual Error: However, upon reviewing the previous steps, we realize that there has been a conceptual error. The simplification of the logarithm to $1$ only applies when the argument of the logarithm is exactly equal to the base, which is not the case here. The argument is a fraction, not just $(x + \frac{1}{2})$ . Therefore, we cannot simplify the logarithm to $1$ , and the previous step is incorrect.

More problems from Quotient property of logarithms

Question

Solve. Write your answer as an integer or a fraction in simplest form.

\newline

5^x=1

\newline

x=

Posted 2 months ago

Question

Steven opened a savings account and deposited

\$100.00

as principal. The account earns

9\%

interest, compounded annually. What is the balance after

5

\newline

Use the formula

A = P(1 + \frac{r}{n})^{nt}

A

is the balance (final amount),

P

is the principal (starting amount),

r

is the interest rate expressed as a decimal,

n

is the number of times per year that the interest is compounded, and

t

is the time in years.

\newline

Round your answer to the nearest cent.

\newline

\$

Posted 2 months ago

Question

Use the following function rule to find

f(1)

\newline

f(x) = 12(7)^x + 8

\newline

f(1) =

Posted 1 month ago

Question

The town of Valley View conducted a census this year, which showed that it has a population of

1,800

people. Based on the census data, it is estimated that the population of Valley View will grow by

12\%

\newline

Write an exponential equation in the form

y = a(b)^x

that can model the town population,

y

x

decades after this census was taken.

\newline

Use whole numbers, decimals, or simplified fractions for the values of

a

b

\newline

y =

Posted 2 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = 9^x

\newline

x =

Posted 2 months ago

Question

Solve. Round your answer to the nearest thousandth.

\newline

7 = e^x

\newline

x =

Posted 1 month ago

Question

Solve. Simplify your answer.

\newline

\log u = 1

\newline

u =

Posted 2 months ago

Question

Solve. Simplify your answer.

\newline

7\log x = 7

\newline

x =

Posted 1 month ago

Question

g(t) = 3^t

change over the interval from

t = 3

t = 4

\newline

\newline

g(t)

3

\newline

g(t)

3

\newline

g(t)

3\%

\newline

g(t)

200\%

Posted 2 months ago

Question

f(x)

6

over every unit interval in

x

f(0) = 0

\newline

Which could be a function rule for

f(x)

\newline

\newline

f(x) = 6^x

\newline

f(x) = 6x

\newline

f(x) = \frac{1}{6^x}

\newline

f(x) = \frac{x}{6}

Posted 2 months ago