# How to solve this logarithm system?

I am new to logarithms and I am having trouble with this logarithm system. \begin{align*} \log_9(x) + \log_y(8) & = 2, \\ \log_x(9) + \log_8(y) & = 8/3. \end{align*} A step-by-step procedure would be highly appreciated.

-
You haven't even had time to read the answer to your other logarithm question. Slow down, digest an answer before posing a new question. – Gerry Myerson Jul 13 '12 at 1:22
Please see here for how to typeset common math expressions with LaTeX. Here's a helpful trick: if you see a math expression on this site for which you want to know the LaTeX code, you can right click on it, go to "Show Math As", then choose "TeX Commands". – Zev Chonoles Jul 13 '12 at 1:23
Use the fact that $$\log_b(a) = \dfrac1{\log_a(b)}$$
Hence, if we denote $\log_9(x) = a$ and $\log_y(8) = b$, we get that \begin{align} a+b & = 2\\ \dfrac1a + \dfrac1b & = \dfrac83 \implies \dfrac{a+b}{ab} = \dfrac83 \implies ab = \dfrac34 \end{align} Now solve for $a$ and $b$ and hence $x$ and $y$.
If $\log(\cdot)$ denotes logarithm in base $10,$ then $$\log_y(x) = \frac{\log(x)}{\log(y)}.$$ Apply through & simplify to see what system you will get. Substitute $X$ for $\log(x),$ and $Y$ for $\log(y).$ You should have 2 equations in 2 unknowns.
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "url": "http://math.stackexchange.com/questions/170130/how-to-solve-this-logarithm-system", "fetch_time": 1462425929000000000, "content_mime_type": "text/html", "warc_filename": "crawl-data/CC-MAIN-2016-18/segments/1461860125897.19/warc/CC-MAIN-20160428161525-00096-ip-10-239-7-51.ec2.internal.warc.gz", "warc_record_offset": 196902519, "warc_record_length": 17243, "token_count": 417, "char_count": 1274, "score": 4.5625, "int_score": 5, "crawl": "CC-MAIN-2016-18", "snapshot_type": "longest", "language": "en", "language_score": 0.826923131942749, "file_path": "/sailhome/yjruan/project/latent-lm/data/processed/finemath-4plus/finemath-4plus/train-00026-of-00064.parquet"}