Solve 2255: Log For All Values Of X3 Write Ln

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Solve the following mathematical problems involving logarithms, exponential functions, and algebraic expressions:

1. Solve the equation (4 / x) = 2 for all values of x.
2. Solve the equation ( ) = - x x Log for all values of x.
3. Express ln(y x) as the sum or difference of logarithms.
4. Solve the equation 2 ln ln( ) + - = + x x ln 5 for all values of x.
5. Express 7 ln x - 3(ln 5 + x ln x) as a single logarithm.
6. Translate log 26.3 = 1.42 into its equivalent exponential form.
7. Translate 4^5 = 625 into its equivalent exponential form.
8. Identify the type of function represented by f(x) = x^6.

Paper For Above instruction

The collection of mathematical problems presented involves a variety of concepts primarily centered around logarithmic and exponential functions, as well as algebra involving powers and polynomial expressions. Addressing these problems requires a solid understanding of properties of logarithms, the ability to manipulate equations, and comprehension of the nature of polynomial functions.

Problem 1: Solving (4 / x) = 2

To solve the equation (4 / x) = 2, we start by isolating x. Multiplying both sides by x gives 4 = 2x. Dividing both sides by 2 yields x = 2. Since division by zero is undefined, x ≠ 0, but in this case, x = 2 is valid, so the solution set is {2}.

Problem 2: Solving ( ) = - x x Log for all x

This problem appears incomplete as presented. Assuming it involves solving an equation of the form log_x(-x) = a or similar, more context is needed. Without additional details, it cannot be accurately addressed. If the intention was to solve an equation involving logarithms with variable bases or arguments, the general approach involves rewriting the equations in exponential form and considering the domains.

Problem 3: Express ln(y x) as the sum or differences of logarithms

Using the properties of logarithms, specifically the product rule, we can express the natural logarithm of a product as the sum:

ln(y x) = ln y + ln x

Similarly, if the logarithm involves a quotient, the difference rule applies:

ln(y / x) = ln y - ln x

Problem 4: Solve 2 ln ln( ) + - = + x x ln 5 for all x

This problem appears complex and possibly contains typographical errors. Interpreting it as an equation involving nested logarithms, perhaps of the form 2 ln (ln x) = ln 5, the solution involves isolating x:

Set 2 ln (ln x) = ln 5. Dividing both sides by 2: ln (ln x) = (1/2) ln 5 = ln 5^{1/2} = ln √5. Exponentiating both sides: ln x = √5. So, x = e^{√5}.

This process demonstrates solving equations involving nested logarithms by employing properties of logarithms and exponentials, while considering the domain restrictions (x > 1 for ln x to be defined).

Problem 5: Express 7 ln x - 3(ln 5 + x ln x) as a single logarithm

Distribute the -3 across the parentheses:

7 ln x - 3 ln 5 - 3 x ln x

Observe that 7 ln x and -3 x ln x are not directly combinable as single logarithms unless rewritten. To express as a single logarithm, note that:

7 ln x = ln x^7

-3 ln 5 = ln 5^{-3}

-3 x ln x cannot be directly expressed as a single logarithm unless the expression is redefined or interpreted differently; perhaps the intended expression is:

ln x^7 - ln 5^{3} - 3 x ln x

Alternatively, if the term involves a multiplicative combination, more clarification is needed. With provided data, the simplest expression is:

ln x^7 - ln 5^{3} = ln (x^7 / 125)

Problem 6: Translate log 26.3 = 1.42 into exponential form

The logarithmic form log_b A = C translates to the exponential form: A = b^C. Assuming base 10 (common logarithm):

26.3 = 10^{1.42}

which can be evaluated as 10^{1.42} ≈ 10^{1} × 10^{0.42} ≈ 10 × 2.63 ≈ 26.3, confirming the equivalence.

Problem 7: Translate 4^5 = 625 into exponential form

This is already in exponential form, with base 4 and exponent 5. To express it in logarithmic form:

log_4 625 = 5

Alternatively, since 4^5 = 1024, but here it equals 625, which indicates an inconsistency. If the intended equation is 4^4.95 ≈ 625, or perhaps 4^4 = 256, 4^5 = 1024. To match 625, the correct base power is not 4; perhaps the problem was to find the logarithm:

log_4 625, which requires changing bases if necessary, or recognizing that 625 = 5^4.

Problem 8: What type of function is f(x) = x^6?

The function f(x) = x^6 is a polynomial function of degree 6. Since the degree is even and the leading coefficient is positive, it is an even degree, even-end behavior polynomial. It is classified as a polynomial function, specifically a power function of even degree, which exhibits symmetry with respect to the y-axis.

Conclusion

These problems highlight the importance of understanding fundamental properties of logarithms and exponents, as well as the structure of polynomial functions. Correctly applying logarithmic rules enables the transformation of complex expressions into manageable forms. Recognizing function types aids in analyzing their behaviors and characteristics, which is essential in advanced mathematics and its applications.

References

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