Questions: Simplify the logarithm 4 log5 x + 3 log5 y - 2 log5 z

Simplify the logarithm 4 log5 x + 3 log5 y - 2 log5 z
Transcript text: Simplify the logarithm $4 \log _{5} x+3 \log _{5} y-2 \log _{5} z$
failed

Solution

failed
failed

Solution Steps

To simplify the given logarithmic expression, we can use the properties of logarithms. Specifically, we can use the power rule, which states that \( a \log_b c = \log_b (c^a) \), and the product and quotient rules, which state that \( \log_b m + \log_b n = \log_b (mn) \) and \( \log_b m - \log_b n = \log_b \left(\frac{m}{n}\right) \), respectively. Applying these rules will allow us to combine the terms into a single logarithm.

Step 1: Apply the Power Rule of Logarithms

The given expression is \(4 \log_{5} x + 3 \log_{5} y - 2 \log_{5} z\). Using the power rule of logarithms, which states that \(a \log_b c = \log_b (c^a)\), we can rewrite each term as:

  • \(4 \log_{5} x = \log_{5} (x^4)\)
  • \(3 \log_{5} y = \log_{5} (y^3)\)
  • \(-2 \log_{5} z = \log_{5} (z^{-2})\)
Step 2: Apply the Product and Quotient Rules of Logarithms

Next, we use the product rule \(\log_b m + \log_b n = \log_b (mn)\) and the quotient rule \(\log_b m - \log_b n = \log_b \left(\frac{m}{n}\right)\) to combine the terms: \[ \log_{5} (x^4) + \log_{5} (y^3) - \log_{5} (z^{-2}) = \log_{5} \left(\frac{x^4 \cdot y^3}{z^2}\right) \]

Final Answer

The simplified expression is: \[ \boxed{\log_{5} \left(\frac{x^4 y^3}{z^2}\right)} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
A company produces foam mattresses in two sizes: regular and king size. It takes 2 minutes to cut the foam for a regular mattress and 6 minutes for a king mattress. If the cutting department has 20 labor-hours available each day, how many regular and king mattresses can be cut in one day? Express your answer as a linear inequality with appropriate non-negative restrictions and draw its graph. Let x be the number of regular mattresses cut per day and let y be the number of king mattresses cut per day. Choose the correct inequality with appropriate non-negative restrictions below. A. 6x + 2y ≥ 1200, x ≥ 0, y ≥ 0 B. 2x + 6y ≤ 20, x ≥ 0, y ≥ 0 C. 2x + 6y ≤ 1200, x ≥ 0, y ≥ 0 D. 6x + 2y ≥ 20, x ≥ 0, y ≥ 0
failedAnswered
Banks create money by making loans and creating deposits, a process that is limited by the size of banks' excess reserves.
failedAnswered
After watching the videos, I found the depiction of 3-D space in both "The Last Supper" and "Night Attack on the SanjS Palace" fascinating. "The Last Supper" by Leonardo da Vinci uses a one-point perspective to create a sense of depth and space. The converging lines lead the viewer's eye to the focal point, creating a feeling of three-dimensional space within the painting. On the other hand, the Sanjo Palace scroll uses a different approach to depict space The sceril presents space as more narrative and panoramic, using layered compositions to convey depth and distance. "The Last Supper" is a mural created with tempera and oil on plaster, while the Sanjo Palace scroll is an ink and color on paper. The medium used in each work contributes to its uniqueness. The mural technique allows da Vinci to achieve high detail and realism. in contrast, the scroil's ink and color on paper technique enables the artist to create a flowing and continuous narrative. In terms of cultural aesthetic preference, both are captivating in their way. The precision and detail in "The Last Supper" appeal to me, as it captures a slgnificant moment in Western religious history with a sense of realism. On the other hand, the dynamic and narrative quality of the Sanjo Palace scroll draws me in due to its depiction of a historical event in Japanese culture. Both cultural aesthetics offer unique perspectives and qualities that make it challenging to choose a preference.
failedAnswered
What is the mass number of an atom with 35 electrons and 38 neutrons? 1.35 2.38 3.70 4.73 5. None of these
failedAnswered
A recent article claims that the state of Illinois has low tuition rates for its colleges and universities. Cate decides to research the cost of tuition for different colleges in Illinois. The average tuition in Illinois is 28,950 with a standard deviation of 3,470. Which type of inference test should Cate use? a) One-sample t-test b) Two-way ANOVA c) One-sample z-test d) Chi-squared test for goodness of fit